Upper Limit of Aerodynamic Forces for Inelastic Wind Design

Tall and slender buildings are sensitive to the dynamic effects of wind flow (commonly known as resonant load) in both service and strength levels. Traditionally, buildings were designed under wind load to behave in the elastic range for both service and strength. Based on the criteria and philosophy, inelastic behavior under service load is considered unacceptable. Meanwhile, no such criterion exists for strength-level load.

To mitigate resonant load through the allowance of plastic deformation, inelastic wind design as part of performance-based wind design (PBWD) has recently received attention. With applied loading being less, design sections are typically smaller, and construction cost can be reduced (El Damatty and Elezaby 2018). Special detailing to safeguard reliable behavior and ensure ductility is essential. Since tall buildings are typically designed for seismic load in addition to wind load, detailing and reliance on seismic available capacity is an option.

The recently published prestandard by ASCE for the PBWD of tall buildings is evidence of the growing importance and acceptance of inelastic wind design among engineers (ASCE 2019). Recommendations to evaluate wind hazards, performance objectives, and corresponding acceptance criteria are presented therein.

The inelastic behavior of structures under wind load has been investigated by many researchers. Inelastic responses of tall buildings using FEM models and nonlinear time-history analysis for wind loads were investigated by Tamura et al. (2001), Bakhshi and Nikbakht (2011), Muthukumar et al. (2013), Judd and Charney (2016), Aswegan et al. (2017), Ghebremariam and Judd (2017), Mohammadi et al. (2019), Chuang and Spence (2020), and Jeong and Kang (2020).

Tamura et al. (2001) studied the inelastic behavior of steel structures of 200 and 300 m height under wind load by performing time-history analysis. The failure mechanism of the structure and the relationship between the level of inelastic deformation of beams and columns and the duration of wind load were examined. In a parametric study by Bakhshi and Nikbakht (2011), responses of steel moment frame buildings with 20, 30, and 40 stories subjected to wind and seismic loads were compared, based on the results of nonlinear time-history analysis. The nonlinear behavior of a 22-story building with reinforced concrete shear wall under wind load was investigated by Muthukumar et al. (2013) to find an appropriate retrofit method. Judd and Charney (2016) studied the inelastic behavior of a 10-story steel frame building by using nonlinear time-history analysis under short and long duration windstorms. Drift and acceleration measured by the analyses were used to evaluate building performance and repair costs, and time was calculated to estimate the fragility of structural and nonstructural components. Aswegan et al. (2017) examined the behavior of a tall building of 66 stories (60 above and 6 below ground level) and a lateral system composed of reinforced concrete core wall and buckling restrained brace (BRB) outriggers under wind and seismic loads. The results of nonlinear time-history analyses revealed that by permitting inelastic wind design, the required section for BRB outriggers could be considerably smaller than that of the design based on elastic design. It was also shown that the use of smaller BRBs results in better performance under seismic load. Ghebremariam and Judd (2017) performed nonlinear time-history analysis for a 46-story building with reinforced concrete shear core walls to evaluate the structural and nonstructural performance level by combining the results of analysis and fragility data. Mohammadi et al. (2019) evaluated the responses of an existing steel moment frame building of 47 stories subjected to wind load through nonlinear dynamic analysis. Component and overall system performance, cladding and interior partitions performance, and comfort performance were investigated as functions of wind speed. Chuang and Spence (2020) proposed a framework for the direct evaluation of post-yield behavior and inelastic deformation at the shakedown state for a structural system discretized by fiber elements under wind load. The accuracy of the method was confirmed by comparing its results with the results of nonlinear time-history analysis. It was shown that the computational efficiency of the proposed method could facilitate stochastic simulation to evaluate the extent and mechanism of failure under severe wind events. Jeong and Kang (2020) carried out nonlinear time-history analysis for a residential building with ordinary reinforced concrete shear walls by using artificially generated time-histories of along-, across-, and torsional-wind loads. The results showed that allowing inelastic deformation to reduce resonant wind force can significantly reduce the amount of reinforcing bars, without sacrificing desirable structural performance.

El Damatty and Elezaby (2018), Elezaby and El Damatty (2020), and Alinejad et al. (2020) also studied the determination of inelastic behavior through elastic analysis, and the application of a response modification factor similar to seismic design. Alinejad et al. (2020) discussed the theoretical aspect, limitation, and framework of PBWD with emphasis on application of the response modification factor for inelastic wind design. Elezaby and El Damatty (2020) attempted to practically implement the concept of the response modification factor for inelastic wind design through the nonlinear time-history analysis of a 65-story building with shear walls as the main lateral resisting system. They found that if the resonant component is reduced by a factor of 2, thickness of walls can be reduced by about 20%–25%. The corresponding performance level for all walls lies between the immediate occupancy (IO) and the life safety (LS) limits.

In evaluating inelastic behavior under wind load, an effective tool is the use of an equivalent single degree of freedom (SDOF) system coupled with nonlinear time-history analysis. In evaluating the failure probability of tall buildings, Vickery (1970) studied inelastic SDOF systems to determine the plastic deformation rate of growth, from which a simple analytical formula was derived to evaluate the mean and standard deviation of inelastic deformation. Chen and Davenport (2000) also investigated the vulnerability of tall buildings under wind load by using inelastic SDOF systems. The results demonstrated that prior to collapse, and in comparison to brittle systems, ductile structures under wind load have considerable reserve safety capacity.

To evaluate damage accumulation under wind load over a period, Hong (2004) studied the mean and standard deviation of inelastic deformation and ductility demand for bilinear inelastic SDOF systems. The results demonstrated that post-yield stiffness can significantly reduce ductility demand. Gani and Légeron (2012) also investigated the relationship between ductility demand and strength reduction by using inelastic SDOF systems under an artificially generated storm and actual measurements for winter storms and hurricanes. Judd and Charney (2015) studied the inelastic behavior of a 10-story building using an SDOF system, performing incremental dynamic analysis under wind load. They examined different types of hysteretic behavior, the effect of cyclic degradation, and the effect of wind duration. Bezabeh et al. (2019) compared the ductility demand of SDOF systems with bilinear self-centering behavior. Here, the effects of wind and structural variables on ductility demand and strength reduction were studied, which included natural frequency, post-yield stiffness, aerodynamic damping, and turbulence intensity. Bezabeh et al. (2020) carried out nonlinear time-history analyses for an SDOF system with different hysteretic behavior, including perfect-plastic, bilinear, pinching, and self-centering, to investigate the behavior of each system under wind load.

In the studies mentioned previously, the potential and mechanism to reduce resonant load through dissipation of kinematic energy by plastic deformation were explored and discussed. Similar to the investigation of inelastic response under seismic load, their main purpose was to define a strength reduction factor (ratio of elastic force to yield strength), and the corresponding ductility demand under wind load.

However, one crucial issue not thoroughly clarified is the limitation regarding aerodynamic force for inelastic wind design. There is no such concern for seismic load because the source of inertial forces is due to excitation at the base, not external forces like wind load. This study is an attempt to investigate the role of aerodynamic force to yield strength ratio in the response of inelastic systems.

This study briefly reviews the concepts of inelastic wind design. Subsequently, the behavior of inelastic systems based on the ratios of aerodynamic force and the yield strength of the system under wind load is discussed. Finally, the topic is investigated by employing inelastic SDOF systems and conducting time-history analysis based on measured aerodynamic forces from the wind tunnel test.

The concept of inelastic design is based on reducing resonant loads by permitting plastic deformation. Consequently, the approach is beneficial where resonant loads are significant, such as for tall and slender buildings. Ideally, if the structural system can mitigate ductility demand in the presence of the resonant force, wind load can be reduced to yield strength. In this case, the aerodynamic force has to be smaller than the yield strength. If the aerodynamic force is larger than the yield strength, wind load can ideally only be reduced to aerodynamic force, not yield strength.

The behavior of inelastic systems where aerodynamic force is larger than yield strength has not been extensively studied. Here, aerodynamic load response is obtained by solving the forced-vibration problem. If the natural frequency of the structure is sufficiently different from the high-energy frequency content of the load, inertial forces are negligible, and the action is static. If not sufficiently different, inertial forces are considerable, and the action is static–dynamic. Static action is tied to the mean and background responses of the load, while the dynamic action is the resonant response. The dynamic action can be interpreted as energy entering into the kinetic system.

For elastic systems, the summation of mean, background, and resonant responses is less than the yield strength. Energy entering the system (through dynamic action) is dissipated through inherent structural damping, with the remaining energy resulting in inertial force. The resonant response due to inertial force is distributed by the structure via internal force and elastic deformation.

For inelastic systems without overstrength (zero post-yield stiffness), two scenarios are considered. The first scenario is where the aerodynamic load, $P_a$, is less than or equal to the yield strength, $P_y$, where the summation of mean, background, and resonant responses is larger than the yield strength [Fig. 1(a)]. Here, part of the energy is dissipated through inherent damping, with the remaining converted to resonant response, and delivered through an elastoplastic mechanism. The resonant response results in inertial force (limited by yield strength), and increases in internal force and displacement (the elastic portion of resonant deformation) and inelastic deformation (the plastic portion of resonant deformation).

In the second scenario, $P_a$ is larger than $P_y$ as shown in Fig. 1(b). For any moment where $P_a$ is less than the yield strength, the mechanisms are similar to the former scenario. However, when $P_a$ exceeds $P_y$, rigid body dynamic motion results due to unstable motion under the $P_a-P_y$ load, since the post-yield stiffness is zero. Under rigid body motion, plastic deformation can dramatically increase and result in structural failure. Deformation also increases due to the increase in entered energy. Similar to other scenarios, part of the energy is dissipated through damping, but the rest is lost through plastic deformation. Thus, ductility demand is much larger than in the case where $P_a$ is less than $P_y$.

In reality, the system has overstrength, and the post-yield stiffness is not zero. Fig. 2 shows that the behavior of these systems is similar to that of systems with zero-post yield stiffness.

Where $P_a$ is equal or smaller than $P_y$ [Fig. 2(a)], the ductility demand is less than that of systems with zero post-yield stiffness. Where $P_a$ is larger than $P_y$ [Fig. 2(b)], no rigid body motion exists, due to available overstrength. Here, internal forces exceed the yield strength to satisfy equilibrium. For systems with small post-yield stiffness, the available ductility can be utilized to satisfy equilibrium.

In this study, a series of numerical simulations was performed to investigate the topic. As a case study, a building of 200 m in height, $H$, (50 stories at 4 meter per story) having a square base plan with a width, $B$, of 40 m was considered. Assuming the following, equivalent single degree of freedom (ESDOF) models were made and analyzed under wind load:

Many empirical equations have been proposed to estimate the natural frequency of a structure, $f$, based on building height. Most of these expressions are biased toward higher estimates of natural frequency for application in seismic design. Goel and Chopra (1997, 1998) proposed lower bound estimates of natural frequency, which are more appropriate for wind design. Based on the commentary in ASCE (2017), this expression is applicable for buildings with a height of less than 122 m. The proposed estimation by Zhou and Kareem (2001) and Zhou et al. (2002) shown in Eq. (1) is for buildings with height taller than 122 mwhere $f$ and $H$ are in Hz and m, respectively. Given the building height ($H=200\mathrm{m}$), the natural frequency for the case study structure is roughly 0.23 Hz.

The stiffness, $K$, of the ESDOF system is assumed to be equal to the generalized stiffness, $K^{*}=M^{*}{(2\pi f)}^2$. Recalling the generalized mass and natural frequency, $K$ is about $297\times {10}^5\mathrm{N}/\mathrm{m}$.

According to ISO (2009), ASCE (2017), and AS/NZS (2011), a damping ratio, $\xi$, of about 1% and 2% may be considered for service- and strength-level wind loads, respectively. At a damping ratio of 2%, the generalized damping coefficient, $C^{*}$, of the ESDOF system was calculated and used, where $2M^{*}(2\pi f)\xi =8.27\times {10}^5\mathrm{N}·\mathrm{s}/\mathrm{m}$. The higher value for $\xi$ was intentionally chosen to demonstrate that even at the higher damping ratio, the issue cannot be restricted.

Aerodynamic wind loads were obtained from the Tokyo Polytechnic University (Tamura 2012) open-access database of wind tunnel tests. The database included results for pressure tests of 1:400 scale rigid models at different aspect ratios, angles of wind direction, and exposure types. The aspect ratios for the case study building were 5:1:1 (height:width:depth), and were assumed to be located at the city center. The results for the model with a mean velocity profile representative of exposure in city centers (power law exponent of 0.25) were extracted. Wind direction normal to the walls was determined to have the largest aerodynamic force. Based on the database, the measured average wind speed at the building specimen height was $11.14\mathrm{m}/\mathrm{s}$, and the total duration and sampling frequency were 32.768 s and 1,000 Hz, respectively. Although the across-wind load (vibration) typically governs the design load for tall and slender buildings, this paper focuses on the relationship between yield strength and aerodynamic force in the along-wind direction, which is also an important aspect of inelastic design.

The time series of local forces were calculated as the product of pressure coefficients and pressure tap tributary areas. The along-wind load at each story was calculated as the summation of local forces on the windward and leeward walls at each story. Since only 25 rows of pressure taps were available for the model (cf. 50 stories for the case study building), each row of pressure taps was considered for two stories. The overturning moment, $M(t)$, was calculated based on the story forces. By assuming a linear mode shape, the generalized load, $P_a^{*}(t)$, was obtained by dividing the overturning moment by the building height, and applied to the ESDOF as an aerodynamic force, i.e., $P_a(t)=P_a^{*}(t)=M(t)/H$.

Two cases of 10-minute average wind speeds with values of 20 and $50\mathrm{m}/\mathrm{s}$ were considered as a representative for low and high wind hazards, which are equivalent to 3-s average wind speeds of 29 and $73\mathrm{m}/\mathrm{s}$, respectively (ISO 2009), for wind speed at building height, $V_H$. Given these wind speeds, a time scale was obtained for each case.

Typically, wind tunnel tests are continuously conducted in the order of 5 to 10 times longer than the target duration (e.g. 10 min or 1 h) to obtain more reliable results and smoother power spectral density (PSD). The challenge was how the time series should be extracted from the results of the wind tunnel tests. Based on the ASCE (2010) recommendations for seismic design, a set of seven time series was derived and applied to the system. It is worth mentioning that in ASCE (2017), the minimum number of time series is increased to 11. However, the ASCE (2010) recommendations were followed for simplification. Table 1 lists, while Fig. 3 shows, the time series obtained by splitting the original load.

Analysis was conducted in two steps. First, the mean value of the load applied to the model was determined through static analysis to eliminate the possible initial impact of the load. Then, the fluctuation component of the load was applied through dynamic time-history analysis for 600 s. The time increments of wind load for 20 and $50\mathrm{m}/\mathrm{s}$ were about 0.2 and 0.08 s, respectively. Based on the primary analysis, it was found that using half of these values (i.e., 0.1 and 0.04 s) was appropriate for performing analysis and extracting outputs.

The yield strength of the ESDOF systems, $P_y$, was defined as a fraction of the maximum aerodynamic force, $P_a$. It was assumed that $\beta P_y$ is equal to $P_a$ of each time series, where $\beta$ was 1.0, 1.25, and 1.5. The post-yield stiffness, $K_{py}$, was defined as $\alpha K$, where $\alpha$ was 0, 0.0025, 0.005, 0.1, and 0.2. The range of $\alpha$ was determined based on the common range used for simplified bilinear systems in FEMA 440 (FEMA 2005). Also, kinematic hardening was adopted for modeling cyclic behavior.

Analyses were carried out to investigate the behavior of inelastic systems subjected to wind load. In the following sections, the results are discussed for the elastic and inelastic systems with zero and nonzero post-yield stiffness. In all, a set of 238 analyses was performed.

The ductility of a structure depends on its structural system, and this value is finite. Alinejad et al. (2020) suggested a target performance for PBWD based on the damage control (DC) level equivalent to the midpoint of immediate occupancy (IO) and life safety (LS) in performance-based seismic design (PBSD). FEMA 356 (FEMA 2000) suggests transient and permanent drift limits of various structural systems by structural performance levels for PBSD. In view of the Alinejad et al. (2020) and FEMA 356 recommendations, the target drift limit of PBWD would be 0.75%, because for high-rise buildings, utilization of a shear wall system is inevitable. Thus, the drift limit of ESDOF is 1.5 m (recalling the 200 m height of the case study buildings).

Fig. 10 shows the acceptance criteria of inelastic behavior under along-wind loads derived from time-history analyses for wind speed of $50\mathrm{m}/\mathrm{s}$. The vertical axis of the graph is the ratio of reduced load, $P$, to the peak aerodynamic force, $P_a$.

If post-yield stiffnesses of 10% and 20% of initial stiffness can be secured, design yield strengths can be reduced to 80% and 67% of aerodynamic forces, respectively. However, it would be more conservative not to reduce the load below the yield strength even for systems with nonzero post-yield stiffness, due to uncertainty in the number of times that peak aerodynamic force can occur. Note that structures designed to have zero post-yield stiffness should have a yield strength equal to, or larger than, aerodynamic forces.

As part of performance-based wind design, inelastic wind design has received more attention as an alternate approach for the design of tall buildings. The reason lies in the large resonant force, which can be interpreted as available potential for dissipation through plastic deformation. The current study recommends limiting the aerodynamic force (summation of mean and background components) to the yield strength of the structure, to reduce the ductility demand and damage accumulation. The results of this study indicate that the limit must be strictly considered for systems with zero post-yield stiffness, to avoid rigid body motion that can result in significant ductility demand and damage accumulation. It is also highly recommended to consider limiting ductility demand for systems with small nonzero post-yield stiffness to a practical range, such that damage accumulation can be reduced, although there is no concern for rigid body motion. For systems with large nonzero post-yielding stiffness—roughly 20% of the initial stiffness—damage accumulation can be significantly alleviated. Given that damage accumulation is considerably increased by the number of times the peak aerodynamic load exceeds yield strength, the recommendation is that in order to achieve a more conservative and safer design, the aerodynamic load be limited to not larger than the yield strength, even for systems with nonzero post-yield stiffness.

All data, models, and code generated or used during the study appear in the published article.

This research was supported by Korea Basic Science Institute (National Research Facilities and Equipment Center) grant funded by the Ministry of Education (No. 2020R1A6C1020263), and by the Engineering Research Institute at Seoul National University. The views expressed are those of the authors, and do not necessarily represent those of the sponsor.