Performance-Based Wind Design of High-Rise Buildings Using Generated Time-History Wind Loads

A virtual 45-story reinforced concrete (RC) building, shown in Figs. 1 and 2, was used to conduct a case study. A building frame system with RC core walls and coupling beams, which are commonly found in lateral load–resisting systems for high-rise buildings, was selected. Building structural information, seismic load, and wind load conditions are summarized in Tables 1–4, respectively.

To determine the inelastic analysis model, initial design is performed per Korea Building Code (MOLIT 2016). Because MOLIT (2016) uses a wind load factor of 1.3 and wind directionality factor of 1.0, the 100-year MRI wind speed in MOLIT (2016) corresponds to that of 1,700-year MRI in ASCE 7-16 (ASCE 2017a). The basic wind speed of $38\mathrm{m}/\mathrm{s}$ used in this paper is also a wind speed for one of the strong wind hazard regions in Korea. Low-to-intermediate seismic and strong wind hazards are assumed, and accordingly, ordinary shear walls and nonconforming transverse reinforcement for beams and coupling beams are used.

To introduce yielding of members under wind loads, reduction of design force is required. Bezabeh et al. (2020) used a load reduction factor for both background and resonant components to compare the damage accumulation of structures with tuned mass damper (TMD) and self-centering systems under along-wind load. However, applying the reduction factor to the resonant component only was recommended for conventional buildings due to large ductility demands. If sufficient postyield stiffness is secured in the conventional buildings, the background component can be reduced. In contrast, El Damatty and Elezaby (2018) proposed the response modification factor ($R_W$) for resonant component only. Because it is difficult to estimate the postyield stiffness of structure in the early stage of design, this approach is employed in this study. The general form of design wind load is per the following equation:where $W$ = design wind load of along-, across-, or torsional-wind load; $W_{\mathrm{mean}}$ = mean component (considered in along-wind load only); $W_{\mathrm{background}}$ = fluctuating of wind load itself; and $W_{\mathrm{resonant}}$ = wind load induced by the resonant effect.

For application, a range for the $R_W$ factor is needed. Design results using an $R_W$ factor of 1, 2, and 3 are compared in this study. $R_W$ factors larger than 3 are not effective for a reduction of the initial design load. Design loads depending on the $R_W$ factor are shown in Fig. 3. An $R_W$ factor of 1 is for conventional elastic wind design. The structural design results of members are shown in the Appendix.

To compare with the seismic load by a response spectrum analysis, the load combination factor of 1.3 was considered for wind loads. The governing lateral design load for high-rise buildings is typically across-wind load due to its large resonant component (Marukawa et al. 1992; Tamura et al. 1996; Ha et al. 2007; Kang et al. 2019). Because large portions of elastic across- and torsional-wind loads are composed of resonant components, those with an $R_W$ factor decrease drastically, whereas the reduction of along-wind is limited due to a large portion of its mean and background components. The reduction of lateral loads significantly affects member forces of coupling beams that mainly resist lateral loads, while the member force reduction of vertical members is insignificant.

There are no current guidelines for inelastic modeling under wind loads. Unlike seismic design, there are few inelastic cyclic loading tests of structural members considering extreme wind loads. Abdullah et al. (2020) studied the inelastic performance of RC coupling beams under extreme wind loads. With hundreds of cyclic loads, RC coupling beams showed a larger pinching effect compared to that by seismic loads, but the overall difference is small. In this study, inelastic behavior of RC structures under seismic and wind loads is assumed to be similar, and PBSD guidelines such as those of the AIK (2019), the Tall Buildings Initiative (TBI 2017), and PEER/ATC 72-1 (PEER/ATC 2010) are employed for inelastic modeling. ETABS version 17 was used for the nonlinear analyses.

ASCE prestandard PBWD (ASCE 2019) permits inelastic behavior under 700–3,000-year MRI wind loads. The performance objective is defined as continuous occupancy.

Acceptance criteria for force-controlled actions are defined as a demand-capacity ratio not exceeding 1.0. The capacity for force-controlled action is the design strength per ACI 318-19 (ACI 2019) with application of appropriate strength reduction factor $\varphi$.

Acceptance criteria for deformation-controlled actions are defined for linear analysis as a demand-capacity ratio not exceeding 1.25. The capacity for deformation-controlled action is expected strength with application of a $\varphi$ factor of 1.0. There are no distinct inelastic behavior capacity criteria of each member for NTHA.

Alinejad et al. (2020) suggested structural performance levels for wind and seismic hazard scenarios based on ASCE 41-17 (ASCE 2017b). Structural performance level up to damage control (DC), which is the midpoint of immediate occupancy (IO) and life safety (LS), was suggested for PBWD. For an upcoming earthquake event, a small $R_W$ factor for PBWD was recommended.

This approach can be employed based on a designer’s judgement for design of buildings—including that for supertall buildings, greater than 200 m per MOLIT (2016), subject to extreme wind hazard and enhanced criteria such as IO.

Along-wind load of KBC is based on the gust load factor (GLF) method suggested by Davenport (1967). The maximum equivalent load is determined by GLF and the standard deviation of the load. The standard deviation can be calculated by the square root of the integration of the PSD function. To calculate the standard deviation of along-wind load, the PSD of wind speed and aerodynamic admittance function were presented. MOLIT (2016) employed the PSD wind speed suggested by Karman (1948). The normalized form of the PSD is shown in Table 5.

To convert wind speed PSD into wind load PSD, aerodynamic admittance needs to be considered. MOLIT (2016) presents the conversion by

By adopting the quasi-steady assumption, the fluctuating component and PSD of along-wind force at the reference height can be expressed by Eqs. (3) and (4), respectivelywhere $t$ = time; $\rho$ = air density ($1.22\mathrm{kg}/{\mathrm{m}}^3$); $A$ = projection area; $v^{\prime }(t)$ = fluctuating component of wind speed; and $C_D^{*}$ = wind force coefficient for fluctuating wind force and can be expressed by (Ha 2017)where $C_D$ = wind force coefficient for mean along-wind force.

Because an enclosed building was assumed, $C_D$ is the difference between the external pressure coefficient of the windward and leeward walls as expressed in Table 6.

The across- and torsional-wind loads in MOLIT (2016) were determined by wind tunnel tests. Empirical equations for PSDs of across- and torsional-wind loads in MOLIT (2016) are shown in Tables 7 and 8, respectively. The PSD of across wind is a form of normalized PSD of overturning moment by the across-wind load, and the PSD of torsional wind is a form of base moment by torsional-wind load.

As shown in Table 8, application of torsional-wind PSD is limited by the range of normalized wind speed, $V_T^{*}$. The range limit of $V_T^{*}$ is assumed sufficient for an estimation of a resonant component with a range of natural frequency common to high-rise buildings. The standard deviation for background and resonant components ${\sigma }_T$ is calculated from the $C_T^{\prime }$ factor rather than the square root of integration of PSD function. But a PSD with sufficient frequency range is required in generation of time-history load. Because the value of $V_H$ used in this study is large, the PSD with $V_T^{*}$ exceeding 10 is assumed in Eq. (6). For the low-frequency range (Fig. 7), it was determined to have a PSD shape similar to the proposed normalized PSD by Kil et al. (2007). Applicable to only certain conditions of the wind speed and size of structure used in this case study is Eq. (6)

Time-history loads are generated from the PSD function as (Hwang et al. 2015)where $S(f)$ = PSD function; $\mathrm{\Delta }f$ = interval of frequency; and $\theta$ = randomly generated phase angle set ($0–2\pi$).

The along-wind time-history load is composed of the mean and background components. Because vertical distribution of the along-wind force is addressed in the wind force coefficient in MOLIT (2016), the along-wind time-history load can be expressed aswhere $z$ = height from the ground; and $X_P$ = generated time history from along-wind force PSD.

The PSDs of across- and torsional-wind loads are moments at the base of the structure. The generated time histories from the base moment PSDs are needed to be converted to the story forces and story torsional moments. In an equivalent static analysis, a vertical load distribution based on a linear mode shape appears appropriate, because across- and torsional-wind loads are governed by the resonant component, i.e., linear first mode shape. However, in the case of time-history analysis, only the mean and background components are applied, and the resonant force is inherently induced during the analysis. For this study, the vertical profiles of fluctuating across- and torsional-wind loads are assumed to be uniform (Ryu et al. 2019). By assuming uniform distribution along the height, across-wind story forces and torsional-wind moments can be expressed by Eqs. (9) and (10), respectivelywhere $z_i$ and $h_i$ = height from ground and story height of the $i$th floor, respectively; and $X_L$ and $X_T$ = generated time histories from across- and torsional-wind PSDs, respectively.

Generated time-history loads have values at the initial start. If the loads were applied directly to a structure, the dynamic response would be overestimated due to sudden loading. Thus, a filter function, $F^{\prime }(t)$, for gradual loading is required. As shown in Fig. 8, an additional 100 s of gradual loading time for the start and end of the wind load duration of 600 s was considered in this study.

The time-history load for each direction can be expressed bywhere $F(t)$ can be $F_D(z,t)$, $F_L(z,t)$, and $M_T(z,t)$.

In this study, the time step of 0.05 s and a frequency range from 0.00125 ($=1/\mathrm{total}\mathrm{time}\mathrm{duration}$) to 10 Hz (Nyquist frequency corresponding to time step) was used.

The possibility of simultaneous occurrence of maximum loads for along-, across-, and torsional-wind loads is low. Thus, wind load combinations for static analysis given in Tables 9 and 10 are presented in MOLIT (2016). The wind load combination in MOLIT (2016) is similar to that of the AIJ (2015). According to AIJ (2015) commentary, the response correlation is negligible between along-wind load and across-wind load, and between along-wind load and torsional-wind load. Meanwhile, the wind load combination factor for across- and torsional-wind loads has been suggested based on the correlation coefficient.

Unlike equivalent static analysis in design codes, the load combination factor is not required if the time histories are appropriately generated so that maximum values of each directional load do not occur simultaneously. Because time histories for along-, across-, and torsional-wind loads are generated based on the random phase angle $\theta$, proper random phase angle sets for each directional load are required to control the occurrence of the maximum load in each direction in the time domain. Because the correlation between along-wind and other direction loads is negligible according to AIJ (2015), a statistically independent random phase angle set for along-wind ${\theta }_{\mathrm{along}}$ was employed. If it is assumed that the trends of PSD shapes in each frequency for across- and torsional-wind loads are the same, and the same random phase angle set is used, the maximum load for each direction will occur at the same instant. If it is assumed that the trends of PSD shape for across- and torsional-wind load are similar, the magnitude of one directional wind load at the instant of the maximum of the other wind load can be controlled by using the same random phase angle set with an additional phase lag as in the following equation:where ${\theta }_{\mathrm{across}}$ and ${\theta }_{\mathrm{torsion}}$ = sets of random phase angle for across- and torsional-wind load, respectively; and $\kappa$ = wind load combination factor for equivalent static analysis in Table 10.

To compare instants of the maximum value occurrence depending on random phase angle sets, 1,000 wind loads were generated. Figs. 9–11 show ratios to the maximum value for other directional loads at the instant of the maximum of one directional wind. Because the mean values of across- and torsional loads are zero, the ratios are determined from absolute values. Fig. 9 represents the statistically independent random phase angle sets of along-, across-, and torsional-wind loads. Fig. 10 is for the same phase angle set of across- and torsional-wind loads, and Fig. 11 is the case if Eq. (12) is used. The expected ratio (average of 1,000 generated time histories) to the maximum value of along-wind load at the instants of maximum of across- and torsional-wind loads is about 0.7. It is much larger than those of across- and torsional-wind loads due to the mean component of along-wind load (Fig. 9). When statistically independent random phase angle sets are used, the expected ratio of across- and torsional-wind loads is about 0.24. Similar results were observed for the expected ratios of across- and torsional winds when along-wind load is the maximum. When the same phase angle set is used (Fig. 10), the expected ratio is 0.9. Using the proposed method, the expected ratio is 0.5, which is close to the value of wind load combination factor $\kappa$.

Even with the proposed method, the deviation of the ratios is still large due to the difference between PSD shapes of along- and torsional-wind loads. Thus, a desirable wind load set shown in Fig. 12 was selected. The load ratios at the instant of maximum of each directional load are given in Table 11. The maximum of each directional load is close to the MOLIT (2016) equivalent static load without resonant component, and is appropriate to be used for the early stage of PBWD. Note that the maximum value in the generated time history is obtained by the superposition of cosine functions with large range of frequencies. Stochastically, the maximum value of time history can be smaller or larger than the expected value (KBC load).

Nonlinear static analysis or pushover analysis for each model was performed to verify the yield strength, yield displacement, overstrength factor ($\mathrm{\Omega }$), and ductility ($\mu$) of the system. The vertical load distribution for pushover analysis was assumed to be the first mode shape. Fig. 13 shows load-displacement curves and seismic performance by AIK (2019) from the analysis. In the building frame system, the lateral load is mainly resisted by the core walls and coupling beams. Due to the relatively short length of the coupling beams, the coupling beams are subject to larger deformation than the shear walls (Lequesne et al. 2016). Thus, inelastic behavior is concentrated in coupling beams when a core wall system is used.

For whole models, yield and collapse of members first occurred at the coupling beams. In the inelastic modeling of coupling beams, ductility capacity depends on the applied shear ratio. If a coupling beam is designed to have a large moment capacity due to elastic wind design ($R_W=1$), the corresponding applied shear force also increases, which results in a reduction of ductility capacity. Moreover, excessive shear forces can be applied to joints and core walls, which leads to a brittle system. To resist the maximum considered earthquake (MCE) with 2,475-year return period, a system with excessively large strength and low ductility may be inappropriate.

For the aforementioned scenario, improved seismic details such as conforming transverse reinforcement in beams and confinement in shear walls are required during a PBSD. On the other hand, buildings designed with $R_W$ factors of 2 and 3 show relatively low yield strengths and sufficient ductility capacity. The overstrength factor and ductility of each model by pushover analysis are summarized in Table 12.

The overstrength factor for the structural system in elastic seismic design in MOLIT (2016) is 2.5. The ductility of a building with a long period is the same as the response modification factor $R_E$ (Chopra 2017). Considering assumed ductility of the structural system, the design seismic load is reduced by the $R_E$ factor in the design code. However, the design result of the wind load governing case ($R_W=1$) deviates from the initial assumption. In other words, the seismic design for the wind load governing case cannot be guaranteed under MCE due to a lack of ductility. By using a larger $R_W$ factor, the design lateral load becomes close to the design seismic load, and the intended results appear. The wind resistance performance in light of ratcheting and low-cycle fatigue should be verified by NTHA.

The duration of wind load is much longer than that of the seismic load. For wind load, extremely large computation is required for NTHA. Thus, the nonlinear modal time-history analysis, called fast nonlinear analysis (FNA) (Wilson 2002), was employed. Despite a deficiency that geometric nonlinearity cannot be considered, FNA was computationally effective compared to the conventional time integration method.

FNA is applicable primarily to a system with small to moderate inelastic deformation, which is suitable for a PBWD. Unlike seismic design with a large reduction of design force by the $R_E$ factor, the $R_W$ factor is limited due to mean and background components, which results in a limited inelastic behavior. For verification, a modal analysis up to 50 modes by Ritz vector was carried out. The accumulated modal participating mass ratios of 99% for $x$-, $y$-, and $rz$-directions were achieved. A modal damping ratio of 1.2% was used for the NTHA.

Analysis results of each model are shown in Figs. 14–16. The model designed by the $R_W$ factor of 1.0 remained elastic. All vertical members remained elastic in all analysis models. In the models with $R_W$ factors of 2 and 3, plastic hinges occurred in the majority of the coupling beams and several beams for the across-wind direction. Due to mean and background components, the reduction of design member forces by along-wind load was limited, and no inelastic behavior was observed for the along-wind direction. Despite a large reduction in design across-wind load, plastic rotation of coupling beams and beams was negligible (Fig. 14) when compared with acceptance criteria (plastic rotation of 0.5%–0.6%) for IO in PBSD. Because of system redundancy, design strength reduction by the $R_W$ factor of 3 resulted in a propagation of yielded members due to force redistribution rather than an increase in plastic rotations.

Fig. 15 shows story forces at the maximum overturning moment for $x$- and $y$-directions. Although the same time-history loads are used for all models, the instants of maximum overturning moment occurrence are different. Vertical load distributions of governing directions in NTHA are similar to those of code-based loads regardless of occurrence instants, because the maximum loads are governed by the first mode shape. However, as shown in Fig. 15(a), those in the perpendicular direction in NTHA are difficult to predict. The gradients of vertical load distributions of along-wind governing cases in NTHA are steeper than that of code-based load due to the resonant component of the first mode shape, while the along-wind story forces in NTHA are quite smaller than those of the code-based force. This appears to be the difference in application of the simplified equation for calculating the background component of along-wind load in MOLIT (2016).

In contrast, across-wind story forces of NTHA and elastic design were close. The difference of the lower part of across wind was due to uniform load distribution of the background component and influence of high-order modes.

The maximum story displacement and drift are shown in Fig. 16. The model with the $R_W$ factor of 1 shows smaller responses than the elastic design model. The smaller response is attributed to the expected strength of concrete and the resultant expected modulus of elasticity in the NTHA model.

The models designed by $R_W$ factors of 2 and 3 show larger responses than the model with the $R_W$ factor of 1 due to yielding. Like the results of the plastic rotation of hinges, displacement and drift of the model with $R_W$ factors of 2 and 3 were almost the same. All of the nonlinear models satisfied the acceptance criterion of peak drift at the top of $H/300$ ($=600\mathrm{mm}$) in the ASCE prestandard (ASCE 2019).

The prestandard for PBWD (ASCE 2019) mentions that in the performance evaluation of a building, it shall be safe to ratcheting and low-cycle fatigue. The ratcheting is defined as a progressive accumulation of plastic deformation leading to P-delta instability.

Bezabeh et al. (2019) introduced inelastic behavior by reducing the wind force so that the mean wind load would be 0.3 and 0.6 of yield strength. The approach does not consider a quasi-static load of the background component, and may result in underestimation of yield strength. When this approach is used for the initial design, performance evaluation for the damage accumulation by along-wind load is required.

In this study, only the resonant component was reduced, and no inelastic behavior by along-wind was observed due to the redundancy of the system. Unlike an idealized SDOF system, which is commonly used in research, real structures in design practice have large postyield stiffness and overstrength. No significant damage accumulation was expected when using the $R_W$ factor in this design approach.

Fatigue failure can occur with a small plastic deformation due to the long duration of wind load. Low-cycle fatigue depends on the number of cycles and extent of plastic deformation. Considering the fundamental natural frequency and time duration of 600 s, the expected total number of cycles is about 135 times and the corresponding strain limit of rebar is about 0.01 (Alinejad et al. 2020). Given that rebar strain is close to its yield strain of 0.00295, plastic deformation is presumed negligible. Thus, sufficient performance is perceived with regard to low-cycle fatigue.

Performance evaluation using generated time-history wind load can be used for a preliminary PBWD. Also, time-history wind loads from wind tunnel tests can be used for verification, if necessary.