## 2 Structural Model, Wind-Loading Model, and Analysis Procedure  
### Equation of Motion and Instantaneous Loads Eccentricities (A-\(\Delta\) Effect)  
For the analysis, actual buildings whose general dimensions are shown in Figure 1a were modeled as three-degree-of-freedom (3 DOF) systems with generalized properties; the 3 DOF system was composed of a rigid slab supported by frames and walls, as shown in Figure 1b. The structure can experience lateral displacements in the X- and Y-direction (i.e., \(u_{x}\) and \(u_{y}\)), as well as rotational displacement, denoted by \(u_{\theta}\). The equation of motion without damping of the structure under wind loading is expressed in Equation (1), where \(m\) is the mass of the structure, \(r\) is the radius of gyration,  
Figure 1: Structural systems: (**a**) general dimensions of the buildings; (**b**) plan view of the three-degree-of-freedom system; and (**c**) movement of the CM in time \(t\) due to wind loads. \(k_{ij}\) for \(I=x\), \(y\), \(\theta\) and \(j=x\), \(y\), \(\theta\) represents the elements of the stiffness matrix, \(F_{x}(t)\) and \(F_{y}(t)\) are the wind-induced forces acting in the X- and Y-direction, respectively, and \(M(t)\) is the wind-induced torsional moment. The methodology used to simulate the wind-induced forces will be discussed in the next section. A dot on a symbol denotes its temporal derivative (i.e., velocity); and a double-dot on a symbol denotes the second temporal derivative (i.e., acceleration). \[ \begin{bmatrix}m&0&0\\ 0&m&0\\ 0&0&mr^{2}\end{bmatrix}\begin{pmatrix}\ddot{u_{x}}\\ \dot{u_{y}}\\ \ddot{u_{\theta}}\end{pmatrix} + \begin{bmatrix}k_{xx}&k_{xy}&k_{x\theta}\\ k_{yx}&k_{yy}&k_{x\theta}\\ k_{\theta x}&k_{\theta y}&k_{\theta\theta}\end{bmatrix}\begin{pmatrix}u_{x} \\ u_{y}\\ u_{\theta}\end{pmatrix} = \begin{cases}F_{x}(t)\\ F_{y}(t)\\ M(t)\end{cases}  \tag{1}\]  
If Rayleigh damping is considered, Equation (1) can be rewritten as  
\[\left\{ \begin{array}{l}\ddot{u_{x}}\\ \ddot{u_{y}}\\ \ddot{u_{\theta}}\end{array} \right\}+a_{1}\omega_{x}^{2}\left[ \begin{array}{ ccc}a_{0}/\left(a_{1}\omega_{x}^{2}\right)+1&0&c_{y}/r\\ 0&a_{0}/\left(a_{1}\omega_{x}^{2}\right)+\Omega_{y}^{2}&c_{x}\Omega_{y}^{2}/r \\ e_{y}/r&c_{x}\Omega_{y}^{2}/r&a_{0}/\left(a_{1}\omega_{x}^{2}\right)+\Omega_{y} ^{2}\end{array} \right]\left\{ \begin{array}{l}\dot{u_{x}}\\ u_{y}\\ e_{y}/r&\Omega_{y}^{2}e_{x}/r&\Omega_{y}^{2}e_{x}/r&\Omega_{y}^{2}\end{array}  \right]\left\{ \begin{array}{l}u_{x}\\ u_{y}\\ u_{\theta}\end{array} \right\}= \tag{2}\]  
where \(\omega_{x}=\sqrt{k_{xx}/m}\), \(\omega_{y}=\sqrt{k_{yy}/m}\), and \(\omega_{\theta}=\sqrt{k_{\theta\theta}/mr^{2}}\) are the circular frequencies associated with each of the degrees of freedom; \(\Omega_{\theta}=\omega_{\theta}/\omega_{x}\) and \(\Omega_{y}=\omega_{y}/\omega_{x}\) are frequency ratios; \(a_{0}=2\zeta\omega_{x}\omega_{y}/(\omega_{x}+\omega_{y})\) and \(a_{1}=2\zeta/\left(\omega_{x}+\omega_{y}\right)\) are the coefficients obtained from the Rayleigh damping; and \(\zeta\) is the damping ratio. To incorporate the A-\(\Delta\) effect into Equation (2), consider Figure 1c that illustrates the movement of the CM with respect to its original position or the CS due to wind loads. The instantaneous movement of CM leads to instantaneous load eccentricities represented by \(u_{x}\) and \(u_{y}\) at time \(t\) that can be incorporated into the time-dependent stiffness matrix given by [11]:  
\[K= \begin{bmatrix}k_{xx}&0&k_{gx}+k_{xx}u_{y}\\ 0&k_{yy}&k_{gy}-k_{yy}u_{x}\\ k_{\theta x}+k_{xx}u_{y}&k_{by}-k_{yy}u_{x}&k_{\theta\theta}+k_{gx}u_{y}-k_{yy}u _{x}\end{bmatrix} , \tag{3}\]  
If the time-dependent stiffness matrix given in Equation (3) is used in Equation (2), Equation (4) represents coupled nonlinear differential equations that incorporate the A-\(\Delta\) effect. \[\left\{ \begin{array}{l}\ddot{u_{x}}\\ \ddot{u_{y}}\\ \dot{u_{\theta}}\end{array} \right\}+a_{1}\omega_{x}^{2}\left[