8 This paper introduces a methodology for developing collapse fragility curve s for windstorms , with a focus 9 on assessing collapse of transmission towers for hurricane event s. Incremental dynamic analysis (IDA) that 10 incorporates the entire hurricane duration is adopted for collapse modeling. F ragility curve s are cumulative 11 distribution function s of a structural limit state such as collapse capacity, which is designated in this work 12 as the intensity measure associated with the onset of collapse. A suite of selected hurricane wind records 13 are used with IDA to pro pagate uncertainties from wind speeds, directions, and durations to collapse 14 capacit ies. Compared with earthquake engineering methodologies , this work proposes appropriate 15 approaches for scaling of wind records, fitting of IDA curves from simulation data, and parameter 16 estimation of hurricane fragility curves. Fragility curves are appropriate for use both to quantify uncertainty 17 in the context of performance -based wind design and for regional loss assessments. As inelastic 18 deformations are allowed in performance- based wind design, it is useful to develop fragility curves based 19 on nonlinear time history analysis for the entire windstorm duration, which has not been addressed in prior 20 work. 21 Keywords: fragility , collapse, full -duration hurricanes , transmission tower , incremental dynamic analysis 22 2 Introduction 23 As structural engineers are moving towards performance- based wind design, geometric nonlinear and 24 inelastic responses of structur es are increasingly being accounted for in design (ASCE 2023; Barbato et al. 25 2013; Chuang and Spence 2019, 2020) . Nonlinear time history analysis (NLTHA) may be adopt ed to assess 26 structural performance and thus demonstrate that inelastic deformations are controlled. Given that nonlinear 27 dynamic analysis is path dependent, NLTHA may be needed for a long duration to represent a windstorm 28 fully . To facilitate performance -based wind design, fragility curves can be used to estimate the conditional 29 probability of failure given a wind hazard intensity. Fragility curves can be developed through conducting 30 NLTHA for a suite of representative wind event s. However, the re is little prior work on 1) characterizing 31 windstorms with a suite of wind records that can capture the nature of the windstorms for the purpose of 32 performance- based wind design, and 2) predict ing dynamic structural responses subjected to a suite of wind 33 records with long or full windstorm duration s of hours to days (ASCE 2023) . To address the first issue , Du 34 et al. (2023) worked on hurricanes and developed a wind map that can provide a suite of representative 35 hurricane wind speed and direction records for a specific location. This work focuses on the second problem , 36 in which a methodology is developed for creating collapse fragility curves using NLTHA for full -duration 37 hurricanes. E lectrical transmission towers subjected to hurricane wind records selected using the approach 38 presented by Du et al. (2023) are presented as an example application . 39 Fragility curves are used to describe the relationship between the failure probability of a structure and the 40 intensity measure of the applied hazard. To calculate the failure probability of a structure for a certain 41 intensity level of hazard is a standard structural reliability problem, where the uncertainties may come from 42 structural properties and hazards with the same intensity level but other different characteristics. Therefore, 43 each point on a fragility curve can be obtained from a structural reliability analysis. Depending on the 44 uncertainties considered in the limit state function, there are time -invariant reliability and time -variant 45 reliability. Consequently, fragility curves can also be developed based on time -invariant reliability or time - 46 variant reliability. 47 3 For the time -invariant reliability approach, the limit state function can be expressed as 𝑔𝑔(𝑿𝑿), where 𝑿𝑿 is a 48 set of random variables. The fragility is defined as the failure probability of the structural system conditional 49 on the specified intensity measure 𝐼𝐼𝐼𝐼, i.e., 𝑃𝑃(𝑔𝑔(𝑿𝑿)≤0|𝐼𝐼𝐼𝐼). To satisfy the pre condition of time -invariant 50 reliability, one should assume the loads are static or consider the dynamic effects im plicitly through some 51 amplification factors. For example, Ellingwood et al. (2004) and Li (2005) developed hurricane fragility 52 curves for light -frame wood construction using design equations with static wind pressure and a gust -effect 53 factor to account for the maximum dynamic respon se. The conditional failure probabilities that form the 54 fragility curves were obtained by increasing the wind speed in 10 mph increments and repeating the first- 55 order reliability method ( FORM ) at each increment . The results were also used to validate the two - 56 parameter lognormal cumulative distribution function (CDF) model of fragility curves by a series of 57 statistical analyses (Li 2005; Li and Ellingwood 2006) , which shows that the lognormal CDF provides a 58 good model for fragility curves of light-frame wood construction. Shanmugam (2011) also developed 59 fragility curves for roof uplift capacities in low -rise residential construction by using static wind loads , but 60 considered correlated non- normal random variables. Shanmugam (2011) accepted the idea that the fragility 61 can be modeled as a lognormal CDF, so S hanmugam did not use FORM or any other reliability methods to 62 calculate failure probabilities, but combined Monte Carlo simulation and the maximum likelihood method 63 suggested by Shinozuka et al. (2000) to estimate the two parameters of the lognormal CDF. Similarly, 64 Shafieezadeh et al. (2013) and Darestani et al. (2022) assumed static wind load s with gust -effect factors 65 according to ASCE 7 (ASCE 2010, 2016) and developed fragility curves for utility wood poles and 66 transmission towers, respectively. However, the gust -effect factor is developed for structural design and 67 assumes that the structure remains linear elastic. Therefore, the design equations with a gust -effect factor 68 may not be suitable for highly nonlinear collapse failure analysis. 69 For the time-variant reliability approach , the failure event is describ ed with respect to a time domain. For 70 example, a time -variant reliability problem may be defined with a limit state function 𝑔𝑔(𝑿𝑿,𝒀𝒀(𝑡𝑡)), where 71 𝒀𝒀(𝑡𝑡) represents a set of random processes such as time histories of wind speeds and ground motions . For a 72 4 certain time t, 𝒀𝒀(𝑡𝑡) becomes a set of random variables; therefore, the instantaneous failure probability at 73 time t may be obtained through the approaches for calculating time -invariant reliabilities. Traditionally, the 74 failure probability is asked to be estimated ove r the duration of an event , which is the first -passage problem 75 and can be addressed by using a series system approximation after discretiz ing the duration into a series of 76 successive short time intervals or by estimating the mean rate of down- crossings of the random process 77 𝑔𝑔(𝑿𝑿,𝒀𝒀(𝑡𝑡)) below zero (Der Kiureghian 2005) . A more practical time -variant reliability problem may be 78 defined by a limit state function 𝑔𝑔(𝑿𝑿,𝒀𝒀(0)~𝒀𝒀(𝑡𝑡)), which means that the instantaneous failure event at time 79 t depends on the random processes 𝒀𝒀(𝑡𝑡) from time 0 to t. For example, due to the dynamic effects and 80 plastic deformations, the structure ’s response at time t dependents on the force time history from 0 to t; thus, 81 the failure probability at time t dependents on the force time history from 0 to t. The method of series system 82 approximation may be used to solve the problem as was done for the limit state function 𝑔𝑔(𝑿𝑿,𝒀𝒀(𝑡𝑡)); 83 however, the correlation s of failure s between different time interval s during the duration of an event need 84 to be considered (Der Kiureghian 2005; Kim et al. 2013; Kim et al. 2019; Song and Ok 2010) . For the limit 85 state function 𝑔𝑔(𝑿𝑿,𝒀𝒀(0)~𝒀𝒀(𝑡𝑡)), if only the failure probability after a n event (i.e., at the end of the loading 86 time history) is of interest, Monte Carlo simulation may be used to account for the uncertainties from 87 random processes 𝒀𝒀(𝑡𝑡). For fragility development, a set of time history analyses may be run using different 88 𝒀𝒀(𝑡𝑡)’s to propagate the uncertainties from 𝒀𝒀(𝑡𝑡) to structural responses. This is the common way to develop 89 fragilit y curves in earthquake engineering, where the failure probability provided by fragility curves is for 90 seismic event s, and fragility curves are usually developed using nonlinear time history analysis with a suite 91 of ground motions. However, collapse fragilities for full -duration windstorm events such as hurricanes and 92 tornados have not been introduced into wind engineering. This is probably due to the massive computational 93 demand to run nonlinear time history analyses for windstorms , whose durations are much longer than 94 earthquakes . 95 In the context of fragility development for transmission towers under wind load s, researchers attempted to 96 develop analytical fragility curves using static analysis (Cai et al. 2019; Darestani et al. 2022) or dynamic 97 5 analysis with fixed time intervals such as 2 minutes (Ma et al. 2021; Tian et al. 2020) , 5 minutes (Fu et al. 98 2016) , and 10 minutes (Fu et al. 2020) . The fragilities developed for these time intervals cannot represent 99 the fragilities for a whole windstorm event because wind speeds and directions may vary during a storm, 100 and failure of a structure incorporates accumulating phenomena such as yielding and dynamic effects. When 101 using these fragility curves, the duration of a windstorm is discretized into a series of short time intervals 102 (e.g., 2 minutes), and the failure probability is calculated for each time interval independently. However, as 103 discussed before , the failure probabilities for those short time intervals are correlated (Der Kiureghian 2005; 104 Straub et al. 2020) . This correlation is difficult to quantify from the view of time -variant reliability and is 105 not considered by the above authors. Hallowell et al. (2018) used the 1 -hour peak wind record to represent 106 a whole hurricane and tr ied to develop fragility curves for a hurricane event. For this case, except for the 107 problem that the structure may be damaged before the 1 -hour peak wind record, another problem here is 108 that the 1 -hour peak wind record is not necessarily the worst hour within a storm. For example, for a 109 transmission tower the wind direction of the 1- hour peak wind record may be parallel to conductors and 110 thus the wind forces on conductors may be negligible, while there may be a worse hour which has a lower 111 wind speed but a wind direction perpendicular to the conductors . 112 To overcome the above -mentioned limitations of the current fragility development strategies for wind load s, 113 this research tries to develop fragilit ies for full-duration windstorm event s and focuses on collapse of 114 transmission tower s subjected to hurricanes . The limit state function has a form of 𝑔𝑔(𝑿𝑿,𝒀𝒀(0)~𝒀𝒀(𝑡𝑡)) since 115 nonlinear and dynamic effects are included in collapse modeling. Instead of assessing failure probabilities 116 for a series of time intervals within a hurricane, the research tries to estimate the failure probability after a 117 hurricane event so that users can avoid the intractable task of quantifying correlations of failure probabilities 118 in different time intervals . The storm -maximum 3-second gust wind speed is adopted as the intensity 119 measure of hurricanes, and a sui te of hurricane wind speed and direction records are selected to represent 120 the uncertainties in wind loading (Du et al. 2023) . The collapse limit state is considered through incremental 121 dynamic analysis (IDA) adapt ed from earthquake engineering (Vamvatsikos and Cornell 2002) , from which 122 6 the collapse capacity ( i.e., the intensity measure associated with the onset of collapse) is obtained . Prior 123 work in the literature that used IDA for capacity assessment of transmission towers under wind loading 124 includes Banik et al. (2010) , Mara (2013) , and Zhang et al. (2015) ; however, the wind records used for IDA 125 are not for storm events and only have 35-second, 1-min or 5- min time intervals with a constant wind 126 direction, which still have difficulties in dealing with the correlations of failure probabilities for different 127 time intervals over the duration of an event . In this work , IDA is performed with a suite of full -duration 128 hurricane wind records so that the results can be used for developing fragilities for hurricane events . The 129 parameter estimation approaches for fragility curves are introduced for cases where the collapse capacity 130 points for all wind records are completely or partially captured on IDA curves . Finally, a methodology is 131 proposed for developing collapse fragility curves for transmission towers in a region using its specific 132 towers and hurricane wind records. 133 Uncertainties in hurricane wind records 134 It is well- known that hurricane wind records with the same intensity measure may have different patterns 135 or uncertainties in terms of the changing of wind speeds and directions. In fragility analysis, these 136 uncertainties should be modeled and propagated to structural responses. T he record -to-record uncertainties 137 can be considered through Monte Carlo simulation, which requires running IDAs for a suite of hurricane 138 wind records collected for a location of interest . To achieve this goal , Du et al. (2023) used Massachusetts 139 as a testbed, discretized it into a series of grids (Fig. 1), and collected hurricanes wind records for each grid 140 from a 10,000- year synthetic hurricane catalog (Liu 2014) . However, due to the high computational demand, 141 it is intractable to use all the collected wind records (about 200 for each grid, some of which may have 142 duration s on the order of 10 hours ) for IDAs and fragility development. A subset of the collected records is 143 used in this work, while still preserv ing the key uncertainties in the loading. This is achiev ed through 144 clustering of the wind records and selecting several of them from each cluster. Using a neural network 145 autoencoder, Du et al. (2023) first compressed the high- dimensional wind records into low dimensional 146 latent features through a encoder process. T he latent features were then expanded to reconstruct the wind 147 7 records through a decoder process. Training of this neural network wa s done to minimize the difference 148 between the original and the reconstructed wind records. Consequently, the low-dimensional latent features 149 contain the most important information in the wind records. Finally, a k-means clustering algorithm was 150 used on the latent features , through which approximately 1/10 of the wind records were selected in each 151 cluster for fragility development . As such, the number of wind records used to run IDA s for each grid is 152 approximately 20. As an example , 16 selected hurricane wind records from 8 clusters are shown in Fig. 2 153 for Grid 86 whose centroid has a latitude of 41.7 and a longitude of -70.1. Here the wind velocities are 154 resolved into the North and East directions because the wind records have changing wind directions . 155 Specifically, the wind records are time series of wind velocity vectors in 2D with a 10-min time step. 156 Therefore, t he clustering and selection process considers the effects of wind duration s, speed s, and 157 directions , which are all reflect ed in the values of the latent features. In Fig. 2, the wind records selected 158 from the same cluster are shown in the same colo r, and it is seen that wind records within the same cluster 159 have similar characteristics in terms of wind speeds , directions , and durations . In addition, 1- hour ramp- up 160 and ramp- down loading histories are added to the beginning and the end, respectively, of each wind record 161 to avoid an impulse effect due to sudden loading (see Fig. 2) for the nonlinear structural dynamic analyses 162 in the following sections (ASCE 2023) . To better compare wind records from different clusters, wind 163 records in Fig. 2 are put together with their midpoint of the duration occurring at the same time. To facilitate 164 the autoencoder, zero paddings at the beginning and end of each record are used to make all records have 165 the same duration in these plots ; however, these zero paddings are removed in the following nonlinear 166 dynamic analysis . As the hurricane wind records used for IDA have different durations , the uncertainties in 167 hurricane wind durations are accounted for in the developed fragilities. 168 8 169 Fig. 1. Massachusetts is discretized into grids with their centroid s shown and labelled 170 171 Fig. 2. An example o f selected hurricane wind records for a grid 172 Hurricane wind loading on transmission towers 173 The selected hurricane wind records are 10-min mean wind speeds at 10 m height . To calculate the wind 174 loads on transmission towers , the wind field along the towers should be generated , which includes modeling 175 of the atmospheric boundary layer and the fluctuating wind speed s. The wind loading time histories may 176 then be calculated based on the equations in the ASCE 74 design code (ASCE 2020) . 177 Wind field simulation 178 9 As an example, Fig. 3 shows the step plot of a 10-min mean wind speed record at a 10 m height, along with 179 the corresponding wind direction record . Note that the wind direction is clockwise positive from the North 180 direction. Based on the 10-min mean wind speeds at 10 m height, the 10-min mean wind speeds at other 181 heights along the tower are calculated according to the logarithmic law boundary layer model (Simiu et al. 182 1974; Simiu et al. 1976; Simiu and Scanlan 1996) , which is 183 𝑈𝑈(𝑧𝑧)=𝑢𝑢∗ 𝑘𝑘ln𝑧𝑧 𝑧𝑧0 (1) where 𝑈𝑈(𝑧𝑧) is the mean wind speed at the height of 𝑧𝑧 ; 𝑢𝑢∗ is the shear velocity; 𝑘𝑘 = 0.4 is the Von Karman 184 constan t; 𝑧𝑧0 is the roughness length of the ground surfac e. In this research, open terrain with a roughness 185 length 𝑧𝑧0= 0. 03 is assumed. After generating 10 -min mean wind speeds for different heights, the 186 fluctuating wind speeds should be superimposed to the mean wind speeds. Here the spatially correlated 187 fluctuating wind speeds are simulated from the Kaimal spectrum (Kaimal et al. 1972) using the spectral 188 representation method (Deodatis 1996; Shinozuka 1972; Shinozuka and Jan 1972) . The correlations of 189 fluctuating wind speeds at different locations are considered through a coherence function, which is an 190 exponential decay as proposed by Davenport (1961) . Fig. 4 presents the simulated and target spectr a of the 191 fluctuating wind speeds with a good match. The simulation of the fluctuating wind speeds is based on the 192 open- source code developed by Cheynet (2020) . After combining the mean and fluctuating wind speeds at 193 different heights, the temporal- spatial evolution of the hurricane wind field along a transmission tower is 194 obtained as presented in Fig. 5, where the seven wind records are for the seven different heights along a 195 transmission tower as shown in Fig. A.1 of Appendix A (see both the red and blue dots in this figure) . Only 196 the absolute values of the wind speeds are shown in Fig. 5, while the changing of the wind directions are 197 omitted for simplicity of the figure . Note that within each 10- min time interval, the wind direction is 198 assumed to be constant even after adding the fluctuating wind speeds. 199 10 200 Fig. 3. An example of the 10 -min mean wind speeds at 10 m height with the corresponding wind directions 201 202 Fig. 4. Simulated and target spectr a of fluctuating wind speeds 203 204 Fig. 5. Temporal -spatial evolution of synthetic hurricane wind speeds 205 Wind force calculation 206 11 Transmission towers are usually discretized into a series of panels along the height (Fig. 6), with wind 207 forces calculated for each panel separately (Mara 2013) . See Appendix A for details of the tower. 208 Calculating wind forces necessitates the orientation of the tower, which is also the orientation of the 209 conductors. These orientations are clockwise positive from the North direction, and the orientation of the 210 tower in this example is assumed to be 0. This means the conductors and ground wires are running in the 211 North -South direction. For transmission tower s subjected to yawed wind, ASCE 74 (ASCE 2020) gives the 212 following equation for wind force calculation on a lattice pane l 213 𝐹𝐹𝑑𝑑=𝑄𝑄𝐾𝐾𝑧𝑧𝐾𝐾𝑧𝑧𝑧𝑧𝑈𝑈3−𝑠𝑠𝑠𝑠𝑠𝑠2𝐺𝐺𝑧𝑧(1+0.2sin2(2Ψ))�𝐶𝐶𝑓𝑓𝑧𝑧𝐴𝐴𝑚𝑚𝑧𝑧cos2Ψ+𝐶𝐶𝑓𝑓𝑓𝑓𝐴𝐴𝑚𝑚𝑓𝑓sin2Ψ� (2) where 𝑄𝑄 is th e air density coefficient with a recommended value of 0.613 (m/s to Pa , converting wind 214 speeds to pressure ); 𝐾𝐾𝑧𝑧 is the win d pressure exposure coefficient ; 𝐾𝐾𝑧𝑧𝑧𝑧 is the topographic factor ; 𝐺𝐺𝑧𝑧 is the 215 structure gust response factor ; Ψ is the yaw angle as shown in Fig. 7; 𝐶𝐶𝑓𝑓𝑧𝑧 and 𝐶𝐶𝑓𝑓𝑓𝑓 are force coefficients 216 associated with the face of the structure that is perpendicular to the transverse and longitudinal directions, 217 respectively ; and 𝐴𝐴𝑚𝑚𝑧𝑧 and 𝐴𝐴𝑚𝑚𝑓𝑓 are area of all members projected in the face of the structure that is 218 perpendicular to the transverse and longitudinal directions , respectively . For conductors and ground wires 219 subjected to yaw angles, ASCE 74 (ASCE 2020) gives the following equation for calculating wind forces 220 perpendicular to the conductor or ground wire 221 𝐹𝐹=𝑄𝑄𝐾𝐾𝑧𝑧𝐾𝐾𝑧𝑧𝑧𝑧𝑈𝑈3−𝑠𝑠𝑠𝑠𝑠𝑠2𝐺𝐺𝑤𝑤𝐶𝐶𝑓𝑓𝐴𝐴cos2Ψ (3) where 𝐺𝐺𝑤𝑤 is the wire gust response factor; 𝐶𝐶𝑓𝑓 is the force coefficient with a recommended value of 1.0; and 222 𝐴𝐴 is the projected area of the wire (i.e., wind span times the diameter of the wire). As recommended by 223 Mara (Mara 2013) , dynamic wind forces on the tower and wires are calculated using Eqs. ( 2) and ( 3) with 224 the two terms 𝐾𝐾𝑧𝑧𝐾𝐾𝑧𝑧𝑧𝑧𝑈𝑈3−𝑠𝑠𝑠𝑠𝑠𝑠2𝐺𝐺𝑧𝑧 and 𝐾𝐾𝑧𝑧𝐾𝐾𝑧𝑧𝑧𝑧𝑈𝑈3−𝑠𝑠𝑠𝑠𝑠𝑠2𝐺𝐺𝑤𝑤 replaced by the simulated wind speeds at the 225 corresponding height . As an example, Fig. 8 shows the calculated wind force time histories. Since the 226 hurricane winds have time-variant wind directions, the wind forces are resolved into transvers e and 227 longitudinal directions for the ease to apply to the structure . Note that the two additional time histories in 228 12 the transverse direction compared with those in the longitudinal direction are forces from one conductor 229 and one ground wire. 230 231 Fig. 6. An 18-m 115 kV transmission tower divided into panels 232 233 Fig. 7. Yawed wind on a transmission tower 234 235 (a) Forces in the transverse direction 236 13 237 (b) Forces in the longitudinal direction 238 Fig. 8. Calculated w ind forces on the transmission tower 239 Collapse fragility development 240 The collapse fragility curves are developed in this section for transmission tower s under hurricanes . First, 241 the mathematical model for fragility curves is briefly introduced. This section then discussed the details of 242 using IDA to capture collapse capacities of towers subjected to the hurricane wind records selected in 243 Section 2. The parameters of fragility curves are then estimated from the data of collapse capacities using 244 the method of moments or the maximum likelihood method. N onlinear dynamic analysis is done using the 245 OpenSees software (McKenna et al. 2010) with the displacement -based beam element developed in Du and 246 Hajjar (2021a, 2021b) for modeling structures made of steel angles and tees such as the transmission towers, 247 where both material and geometric nonlinearit ies are considered. T he axial -flexural -torsional interaction 248 behavior is modeled for steel angles because they (including equal -leg angles) may experience flexural - 249 torsional buckling under complex loading conditions like combined axial forces and moments (Liu and Hui 250 2008) . The Newmark -beta method is used for the integrator with a time step of 0.05 seconds as suggested 251 by Mara (2013) for transmission towers. The uniaxial Steel01 material in OpenSees is adopted with the 252 nominal yield stress. Residual stress is modeled explicitly by applying the residual stress pattern suggested 253 by Kitipornchai and Lee (1986) to fiber section s. Rayleigh damping is adopted with a 2% damping ratio. 254 Other details of the finite element model in OpenSees can be found in Du and Hajjar (2022) . The analyses 255 for the IDAs were run on the DesignSafe cyberinfrastructure (Rathje et al. 2017) . 256 14 Fragility curve and its parameter estimation 257 For the collapse limit state defined with a limit state function 𝑔𝑔(𝑿𝑿,𝒀𝒀(0)~𝒀𝒀(𝑡𝑡)), if the collapse capacity 258 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠 is defined as the intensity measure associated with the onset of collapse for each sample of 𝒀𝒀(𝑡𝑡), 259 the limit state function can be simplified as 260 𝑔𝑔=𝐼𝐼 𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠�𝑿𝑿,𝒀𝒀(0)~𝒀𝒀(𝑡𝑡)�−𝐼𝐼𝐼𝐼 (4 ) C onsequently, the fragility curve is the CDF of the random variable 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠 . This is because the 261 conditional failure probability can be expressed as 262 𝑃𝑃(𝑔𝑔 ≤ 0|𝐼𝐼 𝐼𝐼)=𝑃𝑃�𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠≤𝐼𝐼𝐼𝐼�𝐼𝐼𝐼𝐼� (5 ) I f it is assumed that 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠 follows a lognormal distribution, then the fragility curve can be described 263 as a lognormal CDF with two parameters, median 𝜃𝜃 and logarithmic standard deviation 𝛽𝛽. As used by many 264 researchers (Ellingwood et al. 2004; Shinozuka et al. 2000) , the fragility is defined as 265 𝐹𝐹(𝐼𝐼𝐼𝐼)=Φ�ln(𝐼𝐼𝐼𝐼𝜃𝜃⁄) 𝛽𝛽� (6) w here Φ denotes the standard normal CDF. The collapse capacity 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓 𝑓𝑓𝑐𝑐 𝑐𝑐𝑠𝑠 𝑠𝑠 is defined for a hurricane 266 event and is obtained from IDA. This fragility can only describe the failure probability after an event instead 267 of at any time during the event. 268 P arameter estimation for the fragility curves involves estimating values of the model parameters 𝜃𝜃 and 𝛽𝛽 269 using the simulated data of the collapse capacity. Here the estimates of parameters 𝜃𝜃 and 𝛽𝛽 are denoted as 270 𝜃𝜃� and 𝛽𝛽̂, respectively. Two methods are widely used for estimating the two parameters of fragility curves. 271 The method of moments assumes that the resulting distribution and the simulated data have the same 272 moments. The maximum likelihood method assumes that the resulting distribution makes the simulated 273 data most probable (Baker 2015). Choosing of the parameter estimation method depends on the 274 characteristics of the simulated data. The method of moments requires data set of the collapse capacity 275 15 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠, where parameters of fragility curves can be estimated from the simulated data by taking 276 logarithms of each IDA curve’s 𝐼𝐼𝐼𝐼 𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 value and calculating their mean and standard deviation (Baker 277 2015; Ibarra and Krawinkler 2005) . 278 In addition to the traditional IDA, Baker (2015) also proposed a truncated IDA method, which means 279 conducting IDA only up to some intensity level 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚. This truncated IDA is used due to some concerns 280 of scaling ground motions to very large 𝐼𝐼𝐼𝐼 levels: first, it is computationally intensive; second, the portions 281 of fragility curves at very large 𝐼𝐼𝐼𝐼 levels are of less interest; third, the accuracy of using scaled ground 282 motions with extreme 𝐼𝐼𝐼𝐼 levels to model the real highly intensive hazards is still questionable (Baker 2015) . 283 Here, similar concerns are also present in this research on hurricanes. Therefore, the truncated IDA is also 284 investigated in this paper, which means the hurricane wind records are scaled only up to a relatively large 285 and reasonable intensity level. If all n hurricane wind records used to run IDA cause collapse before the 286 maximum intensity level, then the method of moments can be adopted for parameter estimation. Otherwise, 287 if there are only m records ( m < n) that cause collapse, the method of moments is no longer suitable and 288 instead the maximum likelihood method presented in Baker (2015) is employed for parameter estimation. 289 Specifically, the likelihood that the data set ( m records cause collapse while ( n-m) records does not) can be 290 observed is shown as follows 291 𝐿𝐿𝐿𝐿𝑘𝑘𝐿𝐿𝐿𝐿𝐿𝐿ℎ𝑜𝑜𝑜𝑜𝑜𝑜 =��𝜑𝜑�ln(𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠,𝑖𝑖𝜃𝜃⁄) 𝛽𝛽�𝑚𝑚 𝑖𝑖=1����1−Φ�ln(𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗𝜃𝜃⁄) 𝛽𝛽��𝑛𝑛−𝑚𝑚 𝑗𝑗=1� (7) where 𝜑𝜑() is the probability density function of the standard normal distribution ; 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠,𝑖𝑖 is the 292 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 value for the ith IDA curve ; 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗 is the maximum intensity level after scaling of the jth wind 293 record in the ( n - m) records that did not cause collapse. Here, 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗 is used because different wind 294 records may be scaled up to different intensity levels, which is discussed in detail in Section 4.2. The two 295 parameters 𝜃𝜃 and 𝛽𝛽 can be evaluated by maximizing the likelihood function in Eq. ( 7) through an 296 16 optimization algorithm. It is easier to maximize the logarithm of the likelihood function with getting a 297 mathematically equivalent result, so the parameters can be estimated through 298 �𝜃𝜃�,𝛽𝛽̂�=argmax 𝜃𝜃,𝛽𝛽��ln𝜑𝜑�ln(𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠,𝑖𝑖𝜃𝜃⁄) 𝛽𝛽��𝑚𝑚 𝑖𝑖=1+��ln�1−Φ�ln(𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗𝜃𝜃⁄) 𝛽𝛽���𝑛𝑛−𝑚𝑚 𝑗𝑗=1 (8) Incremental dynamic analysis 299 In earthquake engineering, IDA is one common approach used to assess various limit states of structures, 300 including global collapse capacity, where a suite of ground motions are scaled and applied to a structure in 301 identifying the 𝐼𝐼𝐼𝐼 associated with the onset of collapse (Vamvatsikos and Cornell 2002) . There are two 302 main differences between IDA in wind engineering and earthquake engineering: one is the presence of 303 mean and fluctuating wind speeds compared to the zero mean stochastic excitations of ground motions; 304 another is the variation of the wind profile along the height of the structure (Mara 2013) . To consider these 305 differences, the 10- min mean wind speed records at 10 m height are first scaled for use in creating an IDA. 306 The boundary layer model is then applied based on the scaled 10- min mean wind speed records at 10 m 307 height to generate mean wind speeds at other heights, while the fluctuating wind speeds are generated and 308 added to the mean wind speeds at different heights. For the scaling of ground motions, even though the 309 efficient hunt -and-fill tracing algorithm (Vamvatsikos and Cornell 2002) has been introduced in prior years, 310 researchers often prefer the simpler but more expensive algorithm of scaling up ground motions by a 311 constant 𝐼𝐼𝐼𝐼 increment (Baker 2015) . Similarly, this work also uses a constant increment of the storm- 312 maximum 10 -min mean wind speed for the scaling of wind records. Note that the mean wind speed records 313 instead of the final records including the fluctuating wind speeds are scaled. This is because the generated 314 fluctuating wind speeds depend on the corresponding mean wind speeds at the same location considering 315 that the spectrum and the coherence function of the fluctuating wind speeds are functions of the mean wind 316 speeds. Scaling of the final wind records including the fluctuating wind speeds may invalidate the Kaimal 317 spectrum and the coherence function of the fluctuating wind speeds. Thus, the 𝐼𝐼𝐼𝐼 (i.e., storm -maximum 318 gust speed) increment is not exactly but close to being constant due to the randomness of the fluctuating 319 17 wind speeds. This also explains why a record -dependent 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗 instead of a constant 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚 appears in 320 Eqs. (7) and (8) . Each scaled record is applied to the tower for nonlinear dynamic analysis. A n example of 321 the applied wind speeds and structural response s is shown in Fig. 9, where the sampling frequency is 100 322 seconds. 323 324 (a) Wind speeds and directions at 10 m height 325 326 (b) Structural responses at the top of the tower 327 Fig. 9. Nonlinear dynamic analysis example 328 For transmission towers, the engineering demand parameter (𝐸𝐸𝐸𝐸𝑃𝑃) is chosen as the peak displacement at 329 the top of the tower. An IDA curve is a plot of 𝐸𝐸𝐸𝐸𝑃𝑃 versus 𝐼𝐼𝐼𝐼 and Fig. 10 shows an example of the IDA 330 curves using 𝐸𝐸𝐸𝐸𝑃𝑃 as the horizontal axis and 𝐼𝐼𝐼𝐼 as the vertical axis. To develop this IDA curve, the storm - 331 maximum 10-min mean wind speed of a hurricane wind record is scaled to be 10 m/s to 55 m/s with a 5 332 m/s increment. A total of 10 nonlinear dynamic analyses were performed, but the one with the most 333 intensive wind record did not converge and is omitted in the figure. Here the Piecewise Cubic Hermite 334 Interpolating Polynomial (PCHIP) (Fritsch and Carlson 1980; Kahaner et al. 1989) is employed to generate 335 the IDA curve from the analysis points. Collapse is captured using the 20% slope criterion adapted from 336 18 the IM-based rule in earthquake engineering (Vamvatsikos and Cornell 2002) . As used in earthquake 337 engineering, t he onset of collapse is defined as the last point on the IDA curve with a tangent slope equal 338 to 20% of the elastic slope (see the star in Fig. 10). 339 340 Fig. 10. An IDA curve generated using PCHIP interpolation and the corresponding collapse capacity point 341 Some prior work related to IDA recommends using spline interpolation to generate IDA curves 342 (Vamvatsikos and Cornell 2002, 2004; Vamvatsikos et al. 2003) . The authors found that the traditional 1D 343 spline interpolation is not a good option to capture the collapse capacity point with the 20% slope criterion 344 for hurricane response. The spline interpolation conducts cubic interpolation to construct piecewise 345 polynomials with continuous second- order derivatives, which can be prone to oscillations and overshoots 346 between data points (MathWorks 2022) . Therefore, the authors propose to use the shape -preserving 347 piecewise cubic interpolation, PCHIP, which only has continuous first -order derivatives and has no 348 overshoots and fewer oscillations if the data points are not smooth. Specifically, the PCHIP interpolant is 349 monotonic for intervals where the original data is monotonic. To demonstrate the superiority of the PCHIP 350 interpolation for generating IDA curves, Fig. 11 (a) and (b) compare the IDA curves and collapse capacity 351 points obtained from the same dataset but with a different number of data points. Eleven data points are 352 generated through IDA. It is seen that when using all 11 data points, the PCHIP interpolation produces 353 almost the same collapse capacity point as using the first 9 data points, while the spline interpolation 354 19 produces a different collapse capacity point. For the spline interpolation case, the polynomials fitted from 355 9 data points and those fitted from 11 data points have significant differences in their first derivatives. If 356 collapse happens between points 8 and 9, points 10 and 11 should be unnecessary. This is also important 357 for a truncated IDA in which fewer data points from analysis are intended to provide similar collapse 358 capacity as a corresponding IDA having more data points, as discussed in Section 4.1. 359 360 (a) Using 9 data points 361 362 (b) Using 11 data points 363 Fig. 11. Comparison of the IDA curves and collapse capacity points from PCHIP and spline interpolations 364 Genera tion of fragility curves 365 20 To consider the uncertainties in hurricane wind speed and direction record s, IDAs are run for a tower with 366 a suite of wind records selected for the location of interes t. If all wind records can produce collapse capacity 367 values, then the method of moments is used for parameter estimation of fragility curves . On the contrary, 368 if some of the records do not make the tower collapse until the maximum intensity level , they cannot 369 produce collapse capacity values but can provide some lower bound s. This is designat ed a truncated IDA 370 and the maximum likelihood method is used for parameter estimation . As an example, Fig. 12(a) illustrates 371 16 IDA curves with 16 collapse capacity points for the 115 kV tower shown in Fig. 6, while the 372 corresponding fragility curve is developed using the method of moments . The result is shown i n Fig. 12(b). 373 Here the 16 hurricane wind records displayed in Fig. 2 are used and the storm -maximum 10 -min mean wind 374 speeds are scaled to be 10 m/s to 55 m/s with a 5 m/s increment . The nonconvergent computation results 375 are not included in the figure. To demonstrate a truncated IDA, results from the highest two intensity levels 376 are neglected for parameter estimation , which means the storm -maximum 10 -min mean wind speed are 377 scaled up to 45 m/s . Consequently, the 16 IDA curves can only produce 11 collapse capacity points as 378 illustrated in Fig. 13(a). The corresponding fragility curve is estimated using the maximum likelihood 379 method as in Eq. ( 8) and the result is shown in Fig. 13(b). It is seen that the truncated IDA can produce 380 relatively accurate fragility curves with lower computational demand. Specifically, for this example the 381 computational demand of a truncated IDA is 20% lower than a traditional IDA. 382 21 383 (a) IDA curves and the corresponding collapse capacity points (red stars) 384 385 (b) Fragility curve estimated using the method of moment and the corresponding simulat ed data (black stars) 386 Fig. 12. Collapse f ragility development using traditional IDA 387 22 388 (a) IDA curves and the corresponding collapse capacity points (red stars) 389 390 (b) Fragility curve estimated using the maximum likelihood meth od with the corresponding simulat ed data (black 391 stars) 392 Fig. 13. Collapse fragility development using truncated IDA 393 The truncated IDA discussed earlier is due to the limit of the maximum intensity level. Sometimes an IDA 394 may be truncated due to using a large increment of the intensity measure and the difficulty to converge the 395 nonlinear dynamic analysis. If the analysis does not converge for some higher 𝐼𝐼𝐼𝐼 levels, the generated peak 396 displacements for these 𝐼𝐼𝐼𝐼 levels should not be included in the IDA curve because they are not reliable. If 397 the converged analyses with lower 𝐼𝐼𝐼𝐼 levels have not caused collapse, then this can be treated as a truncated 398 IDA and all the nonconvergent analyses are ignored. Thus, the maximum likelihood method should be used 399 23 for parameter estimation. For the jth wind record, if the nonconvergent analysis starts from 𝐼𝐼𝐼𝐼𝑘𝑘,𝑗𝑗 in the kth 400 increment of the scaled wind records, then the one step lower value 𝐼𝐼𝐼𝐼𝑘𝑘−1,𝑗𝑗 will be used to replace the 401 𝐼𝐼𝐼𝐼𝑚𝑚𝑐𝑐𝑚𝑚,𝑗𝑗 in Eq s. (7) and (8). Theoretically, more accurate results can be obtained by performing more 402 nonlinear dynamic analyses with 𝐼𝐼𝐼𝐼 levels between 𝐼𝐼𝐼𝐼𝑘𝑘−1,𝑗𝑗 and 𝐼𝐼𝐼𝐼𝑘𝑘,𝑗𝑗; however, using the truncated IDA 403 as discussed here can be an alternative way considering the computational intensity of IDAs. In addition, if 404 one is confident that the nonconvergence of the time integration can represent dynamic collapse, then the 405 smallest nonconvergent intensity measure 𝐼𝐼𝐼𝐼𝑘𝑘,𝑗𝑗 will be a upper bound of 𝐼𝐼𝐼𝐼𝑠𝑠𝑐𝑐𝑓𝑓𝑓𝑓𝑐𝑐𝑐𝑐𝑠𝑠𝑠𝑠 and a better parameter 406 estimation can be achieved by replacing the term �1−Φ�ln(𝐼𝐼𝐼𝐼𝑚𝑚𝑚𝑚𝑚𝑚 ,𝑗𝑗𝜃𝜃⁄) 𝛽𝛽�� in Eq s. (7) and (8) with this new 407 term �Φ�ln(𝐼𝐼𝐼𝐼𝑘𝑘,𝑗𝑗𝜃𝜃⁄) 𝛽𝛽�−Φ�ln(𝐼𝐼𝐼𝐼𝑘𝑘−1,𝑗𝑗𝜃𝜃⁄) 𝛽𝛽��; however, this is not done in this work. These equations can be 408 modified if within the development of a fragility curve some IDAs are truncated due to the limit of the 409 maximum intensity level while some other IDAs are truncated due to nonconvergence. 410 Fragility development for a region 411 Fragility curves of transmission towers are essential for fast regional damage assessment of electrical 412 transmission networks. Given the fact that characteristics of hurricane wind records are site- specific, 413 fragility curves may be developed for towers at different locations. To demonstrate this idea, the geographic 414 information of 115 kV overhead transmission lines in Massachusetts is collected from HIFLD open data 415 and shown in Fig. 14. The same grids in Fig. 1 for hurricane wind records selection are used here to assign 416 the transmission towers along the lines to their corresponding grids. In addition, since t ransmission towers 417 are not axisymmetric, the orientation of a tower also has significant impact s on its collapse capacity and 418 fragility curve. Theoretically, the orientation of towers can be obtained from the geographic data of 419 transmission lines; however, developing fragility curves for all existing orientations is intractable due to 420 the huge amount of computational demand. Thus, in this research towers are assumed to be doubly 421 symmetric and fragility curves are only developed for five orientations, which are 0, π/8, π/4, 3π/8, and π/2 . 422 24 Considering that the detailed information of towers is not publicly available, all towers in this 115 kV 423 network are assumed to be the same as the one shown in Fig. 6, which may be unreasonable for practical 424 applications but can be accepted here for a demonstration of the proposed methodology. To summarize, 425 five fragility curves are developed for the 115 kV towers in each grid, and the selected hurricane wind 426 records for each grid a re employed to run IDAs. As an example, Fig. 15 plots fragility curves for the 115 427 kV towers with different orientations in two different grids, where the difference s between the fragility 428 curves in these two grids are due to the site -specific hurricane wind records. When using the fragility curve s, 429 the location and orientation of a tower should be determined first. A n appropriate fragility curve may then 430 be chosen from the developed fragility dataset. Since only five orientations are considered here, users can 431 choose the fragility curve whose orientation is closest to the real orientation of the tower or apply 432 interpolation techniques . The procedure described in this section is only a methodology, since the towers 433 are only representative and cannot be used directly to assess the region of interest . Although only 115 kV 434 transmission lines in Massachusetts are studied here, fragility curves of towers with other voltage levels 435 can be developed using the same methodology. 436 437 Fig. 14. Geographic information of 115 kV overhead transmission lines in Massachusetts 438 25 439 (a) Grid 86 with a centroid of latitude 41.7 and longitude - 70.1 440 441 (b) Grid 78 with a centroid of latitude 41.7 and longitude - 70.5 442 Fig. 15. Fragility curves for 115 kV towers with different orientations in two grids 443 Conclusions 444 This paper presents a methodology for developing collapse fragility curves for structures subjected to 445 hurricanes and uses electrical transmission towers as an example application . The fragility curve describes 446 the collapse probability of a structure after a hurricane event . Compared to the traditional wind fragilities 447 developed for a fixed time interval , the fragilities for hurricane events can avoid the difficulty of quantifying 448 the correlations of failure probabilities for different time intervals in the duration of an event . Uncertainties 449 26 in hurricane wind speeds , directions , and durations are accounted for by assessing collapse capacities of 450 transmission towers with a suite of hurricane wind records for a specific location. Both traditional IDA and 451 truncated IDA are introduced to capture the collapse capacities with scaling of the hurricane wind records. 452 The m ethod of moments and the maximum likelihood method are adopted for parameter estimation of 453 fragility curves based on the collapse capacity data obtained from IDA. Finally, a procedure for developing 454 a set of fragility curves for a region is proposed and demonstrated with consideration s of site -specific 455 hurricane wind records and tower orientations. Performing IDAs for hurricane events is computationally 456 intensive; however, as demonstrated in this paper, these analyses are feasible even for a large region with 457 the currently available computation power. The methodology presented in this work can help in 458 perform ance-based wind design for structures, and the developed fragilities for transmission towers can 459 facilitate damage assessment of electrical transmission networks. 460 Data availability statement 461 Selected simulation cases including the OpenSees model of the tower studied in this paper are available at 462 https://github.com/xinlong-du/TransTowerFragility . 463 Acknowledgement 464 The material presented in this paper is based upon work supported by National Science Foundation under 465 Grant No. CRISP -1638234 , the American Institute of Steel Construction, the American Iron and Steel 466 Institute, the Metal Building Manufacturers Association, the Steel Deck Institute, the Steel Joist Institute, 467 and Northeastern University. This support is gratefully acknowledged. This work used computational 468 resources provided by the Natural Hazards Engineering Research Infrastructure: Cyberinfrastructure 469 (DesignSafe), which is supported by National Science Foundation Grant No. CMMI -2022469. The authors 470 wish to thank Dr. Weichiang Pang at Clemson University for providing the synthetic hurricane catalog and 471 Robert Nickerson for his assistance with this research . 472 Appendix A . Transmission tower details 473 27 This appendix provides drawings of the 150 kV transmission tower used in this paper (Fig. A.1). Member 474 sizes and design loads are given in Table A .1 and Table A .2, respectively. 475 476 Notes: The red dots represent the heights of the wind record s generated for each panel of the tower . The blue dots 477 represent the heights of the wind record s generated for conductors or ground wires. 478 Fig. A.1. Drawings of a 150 kV tower (units: inches) 479 28 480 Table A.1. Member sizes and materials of the tower in Fig. A.1 481 Member Cross s ection (units: inches) Material 1 𝐿𝐿31 2× 31 2×1 4 A36 2 𝐿𝐿5 × 5 ×3 8 A572, Grade 50 3 𝐿𝐿5 × 5 ×3 8 A572, Grade 50 4 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 5 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 6 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 7 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 8 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 9 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 10 𝐿𝐿3 × 3 ×3 16 A36 11 𝐿𝐿4 × 31 2×1 4 A36 12 𝐿𝐿21 2× 2 ×3 16 A36 13 𝐿𝐿4 × 4 ×1 4 A572, Grade 50 14 𝐿𝐿4 × 31 2×5 16 A572, Grade 50 15 𝐿𝐿4 × 4 ×1 8 A572, Grade 50 16 𝐿𝐿31 2× 3 ×1 4 A572, Grade 50 17 𝐿𝐿4 × 31 2×1 4 A36 18 𝐿𝐿5 × 31 2×5 16 A36 19 𝐿𝐿4 × 4 ×3 8 A572, Grade 50 20 𝐿𝐿4 × 4 ×5 16 A572, Grade 50 21 𝐿𝐿5 × 5 ×5 16 A36 22 𝐿𝐿5 × 3 ×5 16 A36 23 𝐿𝐿21 2× 21 2×3 16 A572, Grade 50 24 𝐿𝐿21 2× 21 2×3 16 A36 25 𝐿𝐿21 2× 21 2×3 16 A36 26 𝐿𝐿3 × 3 ×3 16 A36 27 𝐿𝐿2 × 11 4×3 16 A36 28 𝐿𝐿5 × 31 2×1 4 A36 29 𝐿𝐿31 2× 31 2×1 4 A572, Grade 50 30 𝐿𝐿3 × 21 2×1 4 A572, Grade 50 31 𝐿𝐿31 2× 21 2×1 4 A572, Grade 50 32 𝐿𝐿4 × 3 ×1 4 A572, Grade 50 33 𝐿𝐿4 × 31 2×1 4 A572, Grade 50 34 𝐿𝐿3 × 3 ×3 16 A36 35 𝐿𝐿21 2× 21 2×3 16 A36 36 𝐿𝐿3 × 3 ×3 16 A36 37 𝐿𝐿21 2× 2 ×3 16 A36 29 38 𝐿𝐿13 4× 13 4×1 8 A36 39 𝐿𝐿21 2× 21 2×3 16 A36 40 𝐿𝐿13 4× 13 4×3 16 A36 41 Not a member 42 𝐿𝐿31 2× 3 ×1 4 A572, Grade 50 43 𝐿𝐿6 × 6 ×7 16 A572, Grade 50 44 𝐿𝐿21 2× 21 2×1 4 A572, Grade 50 45 𝐿𝐿6 × 4 ×3 8 A36 46 𝐿𝐿21 2× 2 ×1 4 A572, Grade 50 47 𝐿𝐿4 × 4 ×1 4 A36 48 𝐿𝐿3 × 21 2×1 4 A36 49 𝐿𝐿13 4× 13 4×1 8 A36 50 𝐿𝐿2 × 2 ×3 16 A36 51 𝐿𝐿13 4× 13 4×1 8 A36 52 𝐿𝐿13 4× 13 4×1 8 A36 53 𝐿𝐿31 2× 21 2×1 4 A36 54 𝐿𝐿13 4× 11 4×1 8 A36 55 𝐿𝐿2 × 2 ×1 8 A36 56 𝐿𝐿13 4× 13 4×1 8 A36 57 𝐿𝐿31 2× 21 2×1 4 A36 58 𝐿𝐿2 × 2 ×3 16 A36 59 𝐿𝐿31 2× 21 2×1 4 A36 60 𝐿𝐿2 × 2 ×3 16 A36 61 𝐿𝐿3 × 21 2×1 4 A36 62 𝐿𝐿21 2× 2 ×3 16 A36 a 𝐿𝐿13 4× 11 4×1 8 A36 b 𝐿𝐿13 4× 13 4×1 8 A36 Notes: 482 1. Specifications: ASCE Manual & Report on Engineering Practice – No. 52, “Guide for Design of Steel 483 Transmission Towers”, 1971 (except minimum thickness) 484 2. Material: ASTM A36 and ASTM A572, Grade 50 (USS Ex- Ten 50) 485 3. Some cross sections in Table A.1 are not used in the tower shown in Fig. A .1, but they may be used in the body or 486 leg extensions. These extra cross sections are given for completeness, although the drawings of the body and leg 487 extensions are not included in Fig. A .1. 488 4. Ground wires: 2- 7/16” R.S. Steel Max. Tension 489 5. Conductors: 3- 795 MCM ACSR (26/7) 490 6. Weight span: 4,600’ 491 7. Wind span: 4,600’ with 12° Angle in Line 492 3,600’ with 18° Angle in Line 493 2,600’ with 24° Angle in Line 494 1,600’ with 30° Angle in Line 495 496 Table A.2. Design loads (units: kips) 497 Load case Load condition Type of loading Loading points Total load G1 G2 C1 C2 C3 30 1 Intact Vert. 5.80 5.80 12.55 12.55 12.55 49.25 Trans. 8.14 8.14 13.52 13.52 13.52 56.84 Long. N.A. N.A. N.A. N.A. N.A. N.A. 2 Dead end Vert. 2.90 2.90 6.28 6.28 6.28 24.64 Trans. 4.07 4.07 6.76 6.76 6.76 28.42 Long. 11.97 11.97 20.60 20.60 20.60 85.74 3 3 broken conductors Vert. 5.80 5.80 12.55 12.55 12.55 49.25 Trans. 8.14 8.14 6.76 6.76 6.76 36.56 Long. N.A. N.A. 20.60 20.60 20.60 61.80 4 1 broken ground wire & 2 broken cond. Vert. 5.80 5.80 12.55 12.55 12.55 49.25 Trans. 4.07 8.14 6.76 6.76 13.52 39.25 Long. 11.97 N.A. 20.60 20.60 N.A. 53.17 5 2 broken ground wires & 1 broken cond. Vert. 5.80 5.80 12.55 12.55 12.55 49.25 Trans. 4.07 4.07 6.76 13.52 13.52 41.94 Long. 11.97 11.97 20.60 N.A. N.A. 44.54 6 Heavy vertical Vert. 12.85 12.85 22.03 22.03 22.03 91.79 Trans. N.A. N.A. N.A. N.A. N.A. N.A. Long. N.A. N.A. N.A. N.A. N.A. N.A. Notes: In the fragility development process presented in this paper, the tower is assumed under “Intact” load condition, 498 which means no longitudinal forces are transferred to the tower from the conductors and ground wires. 499 500 To validate the finite element model, the tower is analyzed using OpenSees under load cases 1 and 5. The 501 design vertical/gravity loads are first applied to the tower, and the lateral loads are then increased 502 proportionally to the design lateral loads until failure of the tower. Fig. A.2 shows the force- displacement 503 relationships until failure of the tower, where the horizontal axis is the displacement at the top of the tower 504 and the vertical axis is the ratio of the applied lateral force and the design lateral force . It is seen that the 505 capacity of the tower is about 5% to 7% higher than the design loads. 506 507 (a) Load case 1 (b) Load case 5 508 Fig. A.2. Force -displacement curves under two load cases 509 31