## 1 Introduction  
Wind torsional excitation is caused by an imbalance in the instantaneous pressure distributions on all building surfaces and the eccentricity between the elastic and mass centers [1, 2]. Several studies have shown that wind-induced torsional motion is frequently associated with external visual cues due to the rotation of the building [3]. The authors of [4] state that the torsional velocity causes relative motion of distant objects that becomes visually apparent and creates the visual sensation of a \"swinging horizon\" and emphasizes motion perception [5, 6, 7]. Other studies on the wind-induced torsional motion of structures have focused on the development of methodologies to evaluate the structural response on structures. For example, Ref. [1] developed a lateral-torsional motion methodology due to wind loads for idealized buildings and emphasized the importance of the torsional response through the fact that the torsional motion was discernable at a much lower level of response than the lateral translational motion. Moreover, Ref. [8] indicated that the mass or the rigidity center may significantly affect the torsional responses at the geometric center of the building, and that the torsional motion contributes significantly to the overall lateral-torsional coupled responses of a tall building with eccentricities in the mass and/or the rigidity center. The authors of [9] carried out an experimental study to evaluate wind-induced torsion in low- and medium-height buildings; their results indicated that applying 75% of the full wind load with an equivalent eccentricity of 15% improved torsion evaluation. More recently, Ref. [10] carried out an experimental study of wind-induced shear, bending, and torsional loads on rectangular tall buildings and emphasized that the torsional moment induced by the wind forces on tall buildings with rectangular cross sections must be considered during the analysis and structural design stages. It is noted that wind tunnel experiments are of paramount importance in the evaluation of wind-induced loads. In particular, wind tunnel studies have shown to be an excellent tool to evaluate torsional loads on tall buildings with different heights and aspect ratios. More recently, Ref. [11] studied the torsional response under bidirectional seismic excitations for symmetric and asymmetric linear systems. According to [11], the relative distances between the center of mass (CM) and the center of stiffness (CS) vary with time during the ground motion and the bidirectional seismic excitations lead to torsional responses, and instantaneous load eccentricities occur during the horizontal movement of the CM in the plane that can lead to an extra torsional motion not taken in a count in the seismic codes. The author named this second-order effect the A-\(\Delta\) effect. It is noted that the evaluation of the A-\(\Delta\) effect on systems under dynamic wind loading has not been reported in the literature, although several studies have pointed out the importance of the wind-induced torsional response on different types of systems. Although several studies have provided steps towards an understanding of the importance of the wind-induced response, with emphasis on the torsional response, the evaluation of the impact of the correlation of wind forces and torsional moment, as well as the second-order effect, named the A-\(\Delta\) effect, still requires further investigation to evaluate its impact on the wind-induced response. The main objective of this study is to evaluate the impact of the correlation of wind forces, torsional moment, and the A-\(\Delta\) effect on the wind-induced response of two-way asymmetric systems. For the modeling of the structural systems and A-\(\Delta\) effect, coupled nonlinear differential equations are assembled and solved by using the state space model.