The numerical model employed in this study is the same as that developed in Li and Hong (2014), which includes both a wind field model and a full track model. The calculated wind fields using this model compare favourably to the reconstructed wind field from H*Wind (Powell et al., 1998). The HURDAT/HURDAT2 best-track dataset (Landsea and Franklin. 2013) was used to develop the track model, the performance of which was validated by comparing the statistics of the key hurricane parameters calculated from simulated tracks to those from the HURDAT dataset. In addition, Li and Hong (2014) concluded that the estimated return period hurricane wind speed contour maps based on the developed track model, adopted wind field model and equations for defining the wind field parameters result in the comparable wind hazard contour maps to those given in Vickery et al. (2009a). The functional form of the empirical track model is essentially the same as that in Vickery et al. (2000), and similar to that in Vickery et al. (2009a). By using the full track model, storm tracks originating in the North Atlantic basin are simulated from genesis to lysis.

Each simulated track contains the basic information, consisting of latitude and longitude of the storm, heading, translation velocity, and central pressure difference of the storm. These key parameters are obtained for each 6 h time interval and are then linearly interpolated to every 15 min. The information from the interpolated tracks is then used to calculate the wind speed at the particular sites. To simplify the calculation for the 100,000 years of hurricanes activities, and since we are interested primarily in design-level events, only those storms within a radius of 250 km from the site are considered. Parameters defining the wind field, such as the radius to the maximum wind speed (R _max) and the Holland parameter B are calculated from the formulas found in the literature (Vickery et al., 2009a; Vickery et al., 2009b). The final samples of the wind speed duration are 10 min mean wind speeds at the height of 10 m. Figure 1 provides results from Li and Hong (2014), which shows the tracks in the Atlantic basin and the 700 years wind speeds on the Gulf and Atlantic coasts of the United States A comparison of the estimated design level wind speeds based on this model and those recommended in ASCE 7–16 will be presented in Analysis of Conditional Duration Statistics.

In this section, the hurricane wind speed at particular sites is developed and presented as a normalized or standardized function of time. Two typical cases of the wind speed time history are considered: 1) when the eyewall passes over a site, leading to a double peak in the wind speed time history at that site, and 2) when the eyewall misses the site, leading to a wind speed time history with a single peak. In other words, a single peak is formed when the closest distance from the center of the storm to the site is greater than R _max; if the distance is less than R _max, two peaks are formed. Examples of data from the simulation are shown in Figure 2. Samples of wind speed duration are obtained from the simulation results for 10 min mean wind speeds greater than a minimum threshold wind speed, V _min = 18 m/s, which is close to 70% of the lowest value of sustained (1 min mean) wind speed for Category I hurricanes (118 km/h), as defined by the National Hurricane Center. The rationale for the 70% value is discussed below.

The lowest wind speed that is of interest in terms of applying a damaging wind load to the structure is defined as the threshold wind speed, V _T,r . This can be expressed in terms of the proportion of the maximum storm wind speed, V _max, by, r = V T , r V max , (1) where r is the ratio of the threshold to maximum wind speed. The choice for the ratio, r, is discussed further below and in reference to Figure 2.

For the double-peak case, the minimum speed between the two peaks is denoted as V _int . The times corresponding to the threshold wind speeds to the left (i.e., earlier) and right (i.e., later) side of the peak wind speed, V _max, are denoted as T _−Th and T _+Th , respectively. The time corresponding to V _max and V _int are denoted as, T _max and T _int , respectively.

In order to develop a statistically based design cyclone, the wind speed time history is normalized, considering both the maximum wind speed and the duration. For the single peak case, the wind speeds are normalized by the maximum wind speed, V _max, such that, V s = V V m a x . (2)

At the threshold wind speed, V _T,r , the normalized speed, V _s = r. The time duration is normalized according to the following equation, such that the normalized time, T _s = ±1 when V _s = r, i.e., T s = T − T m a x | T m a x − T ± T h | . (3)

It should be noted that, since the durations on either side of the maximum wind speed can differ, the normalizations on either side of the maximum are different. The effective total duration of the hurricane, T _tot , occurs over the time that V _s ≥ V _min /V _max.

For the double-peak cases, additional considerations are required. For simplicity in practical situations (such as the development of cycle counts and loads for a low-cycle fatigue test standard), it is useful to map the double-peak cases as equivalent single peak cases. If the V _int (defined in Figure 2) of a double-peak case is lower than threshold wind speed, the double-peak case is divided into two single peak cases. For double-peak cases for which V _int is greater than the threshold wind speed, to obtain the total duration of the event, T _tot , above the minimum wind speed, V _min, the velocity and time can be normalized using Eq. 2 and Eq. 3 provided V _s ≥ V _min /V _max and the T _Th values are obtained for the extreme time values when V _s = r. To obtain the duration of time when V _s ≥ r, the interval of time when V _s ≤ r is removed, as is the case when the statistics of T _Th are considered in the next section.

The threshold wind speed value also needs discussion. In the design cyclone of Jancauskas et al. (1994), r = 0.7. This yields a wind load that is nominally about half the maximum. It was reported by Gavanski et al. (2016), from an analysis of wind tunnel records, that the range from the lower to the upper bounds of the extreme values of pressure coefficient, Cp, for durations of 10 min, i.e., the ratio of the range of peak pressure coefficients associated with 10 min mean wind speeds is about 0.5. Given this possibility, the reasonable range of wind speeds to consider for the peak loads is over the duration when V _s > ∼0.7. This range would also be appropriate for glazing and glass since static fatigue under wind load is controlled by the peak pressures (Gavanski and Kopp, 2011).

To study the effective total duration of the hurricane wind speeds, the design wind speeds for the ASCE seven Risk Category II for three stations, Miami, Galveston, and Charleston, are used as baseline reference wind speeds. The design wind speeds for Risk Category II buildings based ASCE 7–16 are 3 s gust wind speeds of 75 m/s (168 mph) for Miami, 67 m/s (150 mph) for Galveston, and 65 m/s (146 mph) for Charleston. These correspond to 10 min mean wind speed of 51 m/s for Miami, 46 m/s for Galveston and, 45 m/s for Charleston. For the conversion, a factor of 1.47, calculated based on ESDU (Vickery et al., 2006) is used to convert 3 s gust wind speed to 10 min mean wind speed.

There are four target wind speeds that are selected as V _T,r values. They are wind speeds within 5, 10, 20, and 30% of the design wind speeds for each station. In other words, given the design wind speed, the wind speeds considered in this exercise must be greater than 95, 90, 80 and 70% of the design (peak) wind speed, respectively. Consequently, four threshold 10 min mean wind speeds selected for Miami are 49, 46, 41, and 36 m/s, respectively; for Galveston are 43, 41, 37, and 32 m/s, respectively; and for Charleston are 43, 40, 36, and 31 m/s, respectively. The distribution of the time duration for each of these is shown in Figure 3, along with a fit using the GEV distribution. Four different probabilistic distributions, namely the Lognormal distribution, Gamma distribution, Weibull distribution and Generalized extreme value (GEV) distributions, were fit to the empirical Cumulative Distribution Functions (CDF) in Figure 3. In general, the GEV distribution provides the better fit, based on the AIC criterion (Akaike, 1974).

Figure 3 presents the CDFs for the three locations for the range of threshold wind speeds. Several observations can be made. First, the effective duration ranges from close to zero (i.e., a single 10-min segment) to over 30 hours, depending on the location and threshold speed. Clearly, as the threshold wind speed decreases, the duration increases. Interestingly, the duration of winds for a threshold of 95% of the design speed can be up to 8 hours. However, median values are typically about 2 hours or less for the thresholds examined, while 80th percentile values for the 70% threshold speed are in the range of 2 to 3 hours at the three locations. The threshold wind speed equal to 70% of the design wind speed is the primary focus of the remainder of the analysis in order to develop a statistical basis for cyclone design consideration, based on the rationale provided earlier.

For the development of the statistically based design cyclone, the symmetry of duration about the maximum speeds is investigated to see if there is a bias to one side or the other. The relation of the total time duration is defined as ΔT = T _+Th −T_{− Th} , while the time duration prior to the maximum wind speed is defined as ΔT _L = T _max −T_{− Th} . The relationship of these two terms is investigated for each of three stations by adopting the 70% of the design wind speed as the threshold speed (i.e., V _s = 0.7). Using a lower cutoff threshold wind speed of 30 m/s, the correlation coefficient of ΔT _L and ΔT is about 0.92 for Miami, 0.93 for Galveston, and 0.94 for Charleston. This indicates that ΔT and ΔT _L may be adequately described by a linear relationship. The best linear fit between ΔT and ΔT _L for each of three stations is shown in Figure 4, where the coefficient of determination (R ²) for each fit is greater than 0.85, indicating that the linear model performs adequately. The slope of these linear models ranges from 1.87 to 1.97, which is close to 2.0. For both simplicity and practicality, T m a x = Δ T / 2 could be used since, by adopting such an approximation, the mean wind time history becomes symmetric with respect to T m a x . With this, the modeling of the duration of high winds becomes the assignment of a preferred probability distribution model for ΔT and finding the distribution parameters.

As discussed previously, the double peak samples can be reasonably converted into an equivalent single peak sample. The time duration of a single peak wind speed time series can then be normalized by using Eqs. 2, 3. The threshold wind speed of 0.7V _max is used (which results in all samples being greater than 30 m/s. The normalized storm passages can then be fit into one curve. It was found that a fourth-order polynomial could be used to represent V s ( T s ) adequately, provided that three constraints are applied. This results in: V s ( T s ) = a 0 + a 1 T s + a 2 T s 2 + a 3 T s 3 + a 4 T s 4 , (4) where a _i are the parameters estimated via fitting using the least-square method. The three constraints for Eq. 4 are V _s (0) = 1, V _s (−1) = 0.7 and V _s (1) = 0.7. The first constraint results in a ₀ = 1. The summation of the latter two constraints results in a ₂ + a ₄ = −0.3. The subtraction of the latter two constraints results in a ₁ = −a ₃. This reduces the number of variables in Eq. 5 to two, which can be re-expressed as, V s ( T s ) = 1 + a 1 T s + a 2 T s 2 − a 1 T s 3 − ( 0.3 + a 2 ) T s 4 . (5)

The normalized samples for three stations are fit into Eq. 5. The samples, as well as, the fitted curve are plotted in Figure 5. The R ² value for the fit is about 0.97, which indicates the fourth-order polynomial formula is a reasonable model. In Figure 5, the estimated value a ₁ = 0.0038, and a ₂ = −0.46, which indicates that, on average, the normalized duration can be approximated by a quadratic formula. However, Figure 5 also indicates that there are some relatively large variations of the normalized hurricane wind speed duration, as can be seen in the relatively wide band of the normalized curve. To study the uncertainty in a ₁ and a ₂, each curve based on a sample hurricane is fit to Eq. 5, following which, the samples of a ₁ and a ₂ are fit into several probabilistic models, including the Logistic distribution, Normal distribution, GEV distribution, and Gumbel distribution. Based on AIC criterion, it was found that the Logistic distribution fit the samples the best. The Logistic distribution can be expressed as, f ( a i ; μ , s ) = e a i − μ s / ( s ( 1 + e a i − μ s ) 2 ) , (6) where a i is the random variable, µ and s are distribution parameters that are estimated by using the maximum likelihood method in the present study. The histogram of the obtained values of a ₁, a ₂, and the fitted PDFs are shown in Figure 6, illustrating the adequacy of the Logistic distribution fit. Plots of the value of a ₁, a ₂, and its corresponding maximum wind speed of the storm are also presented in Figure 6 for each sampled hurricane. The R ² and correlation coefficients shown in the plots indicate that the coefficient, a ₁, a ₂, and maximum wind speed of the storm could be assumed to be uncorrelated. It should be noted that the number of available samples of a _i decreases as the maximum wind speed of the cyclone increases.

Design wind speeds from ASCE 7–16 for different risk categories for the three selected locations are listed in Table 1, where the design wind speeds are given in terms of the 3 s gust wind speed (in mph). The predicted 700 years return period values of the hurricane wind speeds by using the model in Li and Hong (2014) are also listed in Table 1. In general, the difference between the estimated 700 years return period values of the hurricane wind speed is within about 3% of the design wind speed in ASCE 7–16 for these three stations based on 100,000 years of simulated hurricanes. This implies that the simulated hurricane wind time series are comparable statistically to those used to derive the design wind speeds for ASCE 7–16.

For the statistical analysis of the effective wind duration for a ₁ and a ₂, given a specified maximum wind speed, it is assumed that the samples from each of the simulated hurricanes with the maximum wind speed within +/− 1 m/s of the specified maximum wind speed can be grouped together. For example, samples of the wind duration for peak wind speed within 50 m/s and 52 m/s can be grouped to represent the samples for a specified wind speed of 51 m/s.

For the probabilistic analysis of the wind duration, three specified wind speeds are selected for each of the considered cities to cover the design wind speed for the different risk categories. The samples of wind duration are fit into four different probabilistic distributions, namely the Lognormal, Gamma, Weibull, and GEV distributions. Based on AIC criterion and the maximum likelihood method, the preferred distribution is the GEV distribution. The empirical cumulative probabilistic distribution and fitted GEV distribution are presented Figure 7.

Practically, for the purposes of developing a design cyclone, one may be more interested in the average storm duration, given the maximum wind speed of a storm. For this purpose, the percentiles of the duration of different maximum storm wind speeds are presented in Figure 8. An exponential model is used to fit the 50-percentile of the duration for each of the three stations. It can be seen from the R ² values provided in the figure that the exponential model fits the data well. Considering the range of design wind speeds from Category I to IV, the median values of the duration are about 1.5–2 h for Miami, 2.5–4 h for Galveston, and 2–3 h for Charleston. The 80th percentile of the duration ranges from two to 3.5 h for Miami, 5–7 h for Galveston, and 3–5 h for Charleston. Such long-duration storms would tend to be more destructive for fatigue sensitive structures and structural components due to the greater number of load cycles and the greater likelihood of larger peak pressures.