The storm track model outlined by Vickery and Twisdale (1995a) and Vickery et al. (2000b) is adopted to simulate hurricanes making landfall at a site of interest. In this model, a hurricane risk region is first formed through a circular subregion centered at a location of interest (e.g., building location). Hurricane tracks are subsequently modeled as straight lines crossing the subregion. Within this context, the hurricane lifetime begins when the hurricane center enters the subregion and ends when it leaves the subregion. In this model, the distance vector between the site of interest and the hurricane center, r _s , at any given time t during the hurricane event is defined as: r s t = cos ⁡ θ ⋅ d m i n − sin ⁡ θ R s 2 − d m i n 2 + c ⋅ sin ⁡ θ ⋅ t ⋅ e + − sin ⁡ θ ⋅ d m i n − cos ⁡ θ R s 2 − d m i n 2 + c ⋅ cos ⁡ θ ⋅ t ⋅ n (4) where d _min is the minimum distance between the hurricane center and the site of interest (taken positive if the site of interest sits to the left of the hurricane track and negative otherwise), R _s is the diameter of the subregion centered at the site of interest, θ is the angle between the storm track and the north direction, and e and n are the unit vectors pointing towards East and North.

The parametric model proposed by Jakobsen and Madsen (2004) is adopted to model the hurricane wind velocity field. The choice of this model was made as it represents a parametric solution to the wind field model outlined by Vickery et al. (2000a) that has been carefully validated and used as the basis of the ASCE 7 wind maps. The implementation of this wind field model is coupled with the hurricane track input vector Θ through the initial central pressure difference (Δp ₀) and the radius of the maximum wind (r _M ). In this model, the mean hourly hurricane wind field at 500 m at time t is solved for the tangential and radial velocity components as: v c r , β , t = v M t r ′ − B ⁡ exp 1 − r ′ − B + a 2 r ′ 2 − a r ′ sin ⁡ β e − cos ⁡ β ⋅ n (5) u c r , β , t = K r ∂ v c ∂ r + r ∂ 2 v c ∂ 2 r − K v c r 2 − C d v c 2 h 1 + α M 2 ∂ v c ∂ r + v c r + f cos ⁡ β ⋅ e + sin ⁡ β ⋅ n (6) where v _c is the tangential component of the velocity field; u _c is the radial component of the velocity field; B is the Holland number; h is the boundary layer thickness; f is the Coriolis parameter; (r, β) are the polar coordinates of a reference system centered at the eye of the hurricane with β = 0 when r points in the positive direction of e; C _d ( ∼0.0015) is a drag coefficient related to the boundary layer average velocity; K is the diffusion coefficient; r′ = r/r _M ; v _M is the maximum tangential velocity given by: v M t = λ B Δ p t e ρ a (7) where ρ _a is the air density, e is the base of the natural logarithmic function, and λ is a coefficient related to the advective, diffusive, and frictional drag terms in the momentum equations defined as 1 / ( 1 + α M 2 ) with α _M = 0.364; and a is a coefficient given by: a = f r M 2 v M (8)

Based on Eqs. 5, 6, the wind field vector v _s at time t can be written as: v s r , β , t = v c r , β , t + u c r , β , t + exp − r r G ⋅ c (9) where r _G ( ∼ 500 km) is the environmental length scale defining the extent to which the translation speed of the hurricane, c, decays in the radial direction. Based on the above definitions, the mean hourly wind speed at a location and height of interest can be estimated through the following transformation: v ̄ H t = 0.1171 ⁡ ln H z 0 z 0 z 01 0.0706 ‖ v s ‖ r s ‖ , β s , t ‖ (10) where H is the height of interest height (e.g., building height), z ₀ is the terrain roughness length at the site of interest, z ₀₁ is the roughness length at 10 m in open terrain, 0.1171 is a dimensionless coefficient related to transforming wind speeds from 500 to 10 m in open terrain, and β _s is the angle in polar coordinates between the eye of the storm and the site of interest.

As the hurricane moves along its track, the wind speed, v ̄ H ( t ) , continuously varies due to variations in the wind velocity field and relative position of the hurricane center to the site of interest. The corresponding time-varying wind direction, α _H (t), at the site of interest can be determined from v _s (‖r _s ‖, β _s , t) estimated for the current wind velocity field.

Once hurricanes make landfall, the central pressure difference (Δp) will in general decay resulting in a reduction in the wind field and hence the wind speed at the site of interest. To simulate this phenomenon, the following filling-rate model is adopted (Vickery and Twisdale, 1995b): Δ p t = exp − a f t Δ p 0 (11) where an exponential decay is used to model the dissipation of the hurricane central pressure deficit once landfall is made. To include uncertainties in the decay rate, the following probabilistic filling constant a _f , dependent on the initial central pressure difference Δp ₀, is considered: a f = a 0 + a 1 Δ p 0 + ϵ f (12) where ϵ _f is a zero-mean normally distributed error term with standard deviation σ _ϵ while the parameters a ₁ and a ₂ are site-specific and model the expected decay. Suggested values for various locations for a ₁, a ₂, and σ _ϵ can be found in the study by Vickery and Twisdale (1995b). The parameters a ₀, a ₁, and ϵ _f are included in the hurricane input parameter vector Θ.

To model the concurrent rainfall, the IPET (Interagency Performance Evaluation Task) parametric precipitation model, developed based on the National Aeronautics and Space Administration’s Tropical Rainfall Measuring Mission database, is adopted (Lonfat et al., 2004; Chen et al., 2006). Comparative studies have suggested this model is superior to other commonly used parametric rainfall models (Lonfat et al., 2004; Brackins and Kalyanapu, 2020). From the IPET model, the evolution of the mean hourly horizontal rainfall, R _h (t), can be estimated at the site of interest directly from the hurricane parameters Δp(t), r _s (t), and r _M at any given time, t, through the following expression: R h t = 1.14 + 0.12 Δ p t ; r s t ≤ r M 1.14 + 0.12 Δ p t exp − 0.3 r s t − r M r M ; r s t > r M (13) where Δp is in millibars, R _h is in h/mm, and r _s (t) and r _M are in kilometers. The value calculated by Eq. 13 provides the symmetric component of the rainfall field. To estimate the asymmetric component, R _h (t) can be multiplied by a factor of 1.5 if the site of interest is in the northern hemisphere and to the right of the hurricane track (0.5 if it is to the left).

Based on the straight and stationary wind pressure simulation model outlined by Ouyang and Spence (2020), a non-stationary/-straight wind pressure model is developed to capture the effects on the aerodynamic pressures of the continuously varying wind speed and direction associated with full hurricanes. The main steps of the model are outlined in the conceptual flowchart of Figure 1. The model is calibrated to data in the form of vectors of model-scale surface pressure coefficients C _p,e,M (t _M ), with t _M the model-scale time, collected in wind tunnel tests where stationary/straight but non-Gaussian pressures are measured at a grid of sensors on the model surface for a discrete set of wind directions (e.g., {10°, 20°, …, 360°}). To reconcile the discrete wind directions of the wind tunnel data with the continuously varying wind directions of the hurricane track, these last are transformed into a piece-wise discrete representation, as illustrated in step (I) of Figure 1, where a set of segments with constant wind directions are defined. In step (II), model-scale stationary/straight but non-Gaussian wind pressure coefficient vector processes, C p , e , M ( i ) ( t M ) , are generated for each segment through the straight/stationary but non-Gaussian models outlined in Ouyang and Spence (2020). In step (III), the continuous wind directions are approximated through a piece-wise linear representation to which the segments of straight/stationary and non-Gaussian pressures are merged therefore leading to a non-stationary/-straight/-Gaussian representation of the pressure coefficient vector process, C _p,e,M (t _M ), for the full hurricane event at model-scale. Finally, C _p,e,M (t _M ) is mapped back to the building-scale time in step (IV) and translated to the non-stationary/-straight/-Gaussian process, p _e (t), in step (V).

In the following, further details of each step of the model outlined in Figure 1 are provided.

Based on the results reported by Ouyang and Spence (2021), the structural system is assumed to respond elastically. The dynamic response of the structural system can therefore be estimated through solving the following modal equations: q ̈ i t + 2 ω i ζ i q ̇ i t + ω i 2 q i t = Q i N t (28) where q _i , q ̇ i , and q ̈ i are the displacement, velocity, and acceleration associated with the ith dynamic mode; ω _i and ζ _i are the circular frequency and modal damping ratio of the ith mode, while Q i N ( t ) is the non-stationary/-straight/-Gaussian generalized force of the ith mode estimated as: Q i N t = ϕ i T ϕ i T M ϕ i f ̃ N t (29) where ϕ _i is the ith mode shape; M is the structural mass matrix; and f ̃ N ( t ) is the dynamic forcing vector evaluated through integrating the non-stationary/-straight/-Gaussian pressures of Eq. 25.

From the solution of Eq. 28, the dynamic structural response can be approximated from the first N _m modes as: x t ≈ ∑ i = 1 N m ϕ i q i t (30)

Dynamic story drift, Dr(t), at any location of interest can then be directly estimated through a linear combination of the appropriate components of x(t).

The net pressure demands at an envelope location ξ _xyz of interest, p _n (t, ξ _xyz ), are evaluated as: p n t , ξ x y z = p e t , ξ x y z − p i t , ξ x y z (31) where _e (t, ξ _xyz ) is the external pressure estimated through the models of Section 4.1 at ξ _xyz while p _i (t, ξ _xyz ) is the corresponding internal pressures. To estimate the dynamic internal pressures p _i (t, ξ _xyz ), the interior of the building is modeled as a system of interconnected compartments. Initially, the building is considered enclosed with negligible internal pressurization. During the hurricane, openings can be created in the envelope due to component damages, which allows air to flow into or out of the building triggering dynamic internal pressures in all compartments that are connected through an internal opening. To solve the transient air flows, the internal pressure model outlined by Ouyang and Spence (2019) is adopted, in which the air velocity at each opening is described through the unsteady-isentropic form of the Bernoulli equation (Vickery and Bloxham, 1992; Yu et al., 2008; Guha et al., 2011). To treat the time dependency of v ̄ H , the dynamic internal pressures, p _i (t, ξ _xyz ), at each opening (external/internal or internal/internal), are directly estimated through solving a system of nonlinear equations (one for each opening) derived based on the principle of mass conservation. A fourth-order Runge Kutta scheme can be used to solve the system where, at each time step, the pressure-induced damages are iteratively updated until dynamic equilibrium is achieved.

It is important to observe that in solving for p _i (t, ξ _xyz ), the current drift-induced damage state of each envelope component must be considered. This couples not only the structural and pressure demands (e.g., a drift-induced damage to the envelope can cause air flow therefore affecting the internal pressure) but also the demand and damage analysis (e.g., the occurrence of a drift- or pressure-induced damage state can affect internal pressures). It should also be observed that damage to the envelope is progressive in nature as it accumulates over the duration of the event.

To model the damage susceptibility of the ith envelope component to N D r i drift-induced and N P i pressure-induced damage states, suites of N D r i and N P i sequential damage thresholds are defined: C P i = { C P 1 i ≤ C P 2 i … , ≤ C P N P i } and C D r i = { C D r 1 i ≤ C D r 2 i … , ≤ C D r N D r i } . The randomness in the thresholds is modeled through corresponding suites of sequential fragility functions. At a given time step, t ̂ , all component thresholds are compared with the current story drift demand, D r i ( t ̂ ) , and net pressure demand p n ( t ̂ , ξ x y z i ) , where the largest exceeded threshold defines the current pressure- and/or drift-induced damage state. To model potential coupling between drift- and pressure-induced damage states (e.g., the occurrence of a drift-induced damage state could affect the capacity of the component to resist net pressure and vice versa), the thresholds of a suite of coupled damage states are probabilistically degenerated upon the occurrence of the coupled damage state. The final damage states of each envelope component represent the system measures of interest.

The concurrent rainfall leads to the deposition of rainwater on the envelope. Damage to the envelope can then lead to water ingress. To estimate the volume of water ingress, the flow rate at each opening can be estimated directly from R _wdr (ξ _xyz , t), estimated through the models of Section 4.2, and the steady-state water runoff solution derived by Ouyang and Spence (2019). From the flow rate at each opening, the total volume of water entering through an opening at a given time, t ̂ , can be estimated by integrating the flow rate from the time the opening first occurred, i.e., the time at which the damage causing the opening occurred. Through the implementation of the water ingress model, the time traces of total volume of water entering through each opening can be estimated.

To enable the comparison between the full hurricane model of this work and a classic nominal hurricane setting, for each full hurricane sample, a nominal hurricane is also generated based on the maximum wind speed v ̄ H , with associated direction α _H , and the maximum rainfall intensity to occur over the duration of the full hurricane. For both nominal and full hurricanes, a uniform time step of Δt = 0.5 s at building-scale is used.

To illustrate and discuss the evolution of damage during a full hurricane event, a single hurricane event is analyzed in detail in this section. The event corresponds to a category five hurricane on the Saffir-Simpson scale (Taylor et al., 2010), with a maximum mean hourly wind speed at the building top of 67.7 m/s. The time evolution of mean hourly wind speed v ̄ H ( t ) , wind direction α _H (t) (measured counterclockwise from south), and mean hourly rainfall intensity R _h (t) is reported in Figure 4. An example of the corresponding non-stationary/-straight/-Gaussian wind pressure simulated through the procedure of Section 4.1 is shown in Figure 5 for an envelope component located at the upper-left corner of the south face of the building.

Figure 6 reports the accumulation of damage over the duration of the hurricane in terms of the total number of envelope components assuming D S D r 1 , D S D r 2 , or D S P 60 . From the comparison between the damage histories and the wind speed history of Figure 4A, it can be seen that most damage occurs near the time of the maximum wind speed, i.e., during the 7th hour of the hurricane event. By the end of the hurricane event, the final damage states for each envelope component were recorded and are reported in Table 2 in terms of the number of damaged components on each face of the building. As can been seen, due to the continuously varying wind direction, the damage is relatively evenly distributed between the faces. The distribution of final damages shows how pressure-induced damages are dominant, which is consistent with the results reported by Ouyang and Spence (2020) for a nominal hurricane representation. Water ingress is also recorded during and at the end of the hurricane, where a total volume of 270.5 m³ of water was estimated to enter the building through the damaged envelope components. The time histories of water ingress at each floor during the hurricane are reported in Figure 7 and show how water ingress towards the bottom of the building dominates.

The mean annual rate of each envelope component assuming as a final damage state D S D r 1 , D S D r 2 , or D S P 60 is reported in Figure 8. The damage maps show how the drift-induced damages are uniformly distributed over the envelope except for the top and bottom floors, while the pressure-induced damages are more concentrated near the edges of the building due to the local aerodynamic response of the system. Overall, the damage patterns and rates are similar to those seen for the nominal hurricane setting analyzed by Ouyang and Spence (2020).

To evaluate the system-level envelope performance for both the nominal and full hurricanes, Figure 9 reports the damage curves for both scenarios in terms of the mean annual rate of exceeding a total number of components assuming as a final damage state D S D r 1 , D S D r 2 , or D S P 60 . Comparison between the drift-induced damage curves shows how the total number of damaged components is well estimated by the nominal hurricane for annual rates greater than 2 × 10^–6. However, for rarer events, the nominal hurricane will generally lead to considerable overestimation of damage. For pressure-induced damage, it can be seen that the nominal hurricanes underestimate the damages for mean annual rates greater than 2 × 10^–6, but once again significantly overestimate the damages for rarer events. The differences in Figure 9 are likely caused by the duration of the maximum wind T _m , where T _m is defined as the duration when the hurricane wind speed v ̄ H ( t ) is within a certain percentage of the maximum wind speed v ̄ ̂ H = max [ v ̄ H ( t ) ] (e.g., v ̄ H ( t ) ≥ 0.95 v ̄ ̂ H ). Indeed, the storm track model considered in this study suggests that hurricanes with a larger maximum mean wind speed, v ̄ ̂ H , have a relatively “sharper” wind speed history curve (i.e., the duration of the maximum wind is shorter).

To investigate this, the distribution of maximum wind speed duration is analyzed for all hurricane samples in hazard intervals three to nine, where the first two intervals are not considered as the value of v ̄ ̂ H is negligible from an engineering standpoint. The mean and standard deviation of the durations are reported in Figure 10, from which it can be seen that as the hurricane event becomes rarer, the duration of maximum wind becomes shorter. In particular, it can be seen that wind speeds within 98% of the maximum have an expected duration of around one-hour duration. The capability of the nominal hurricane to adequately reproduce the damage would suggest that envelope damage is occurring essentially when wind speeds are at their maximum.

The loss curves associated with repair costs are reported in Figure 11. The relative magnitude of total repair cost between the nominal and full hurricanes is similar to the damage curves of Figure 9C, which implies that the pressure-induced damages dominate the total repair cost associated with the envelope components. Figure 12 reports the exceedance rates associated with the consequence metric of total volume of water ingress V _W . From the comparison of the water ingress curves, the nominal hurricane significantly underestimates the total amount of water ingress as compared to the full hurricane. To quantify this underestimation, Table 3 reports the total water ingress at different exceedance rates for the nominal and full hurricanes. As can be seen, a near 40-fold underestimation of water ingress can be seen for exceedance rates of 1 × 10^–3. The root of this difference can be traced back to how the nominal hurricane neglects the water that can enter the building due to rainfall after the peak wind speeds have occurred. As the exceedance rates decrease, the underestimation of total water ingress from the nominal hurricane also decreases. This is due to how as the hurricane events become more extreme, the majority of damage will occur at the beginning of the nominal hurricane event therefore increasing the duration in which water can ingress.