## 2 Continuous Shear-Flexural Beam Model  
Despite the complex analysis of multi-degree-of-freedom (discrete mass) systems, continuous models are widely used in the literature as approximate methods. In recent years, various continuous beam models and simplified versions based on beam theories have been proposed to predict structural responses. Typically, these simplified models consider a continuous cantilever beam with uniform mass and stiffness distributions along its height. In these studies, displacement and rotation are taken to be zero as the base boundary condition. This approach does not fully represent the behavior of the structure since rotation gives drift in the continuous model. This study aims to enhance the continuous system model approach and improve predictions of structural responses by introducing a closed-form solution that incorporates a rotation (\(\theta\)) boundary condition at the base, as utilized by [PERSON] and [PERSON] [5] in their continuous shear-flexural beam model with fixed base. When subjected to a horizontal ground acceleration \(\dot{\upmu}_{g}(t)\) at the base, the undamped motion equation of the continuous beam model is formulated as a partial derivative differential equation, represented as Equation (1). \[\frac{m}{EI}\frac{\partial^{2}u(\zeta,t)}{\partial t^{2}}+\frac{1}{H^{4}}\frac {\partial^{4}u(\zeta,t)}{\partial\zeta^{4}}-\frac{\alpha^{2}}{H^{4}}\frac{ \partial^{4}u(\zeta,t)}{\partial\zeta^{4}}=-\frac{m}{EI}\frac{\partial^{2}u_{g }(t)}{\partial t^{2}} \tag{1}\]  
where m is the mass per unit length, H is the total height of the building, u(\(\zeta\),t) is the lateral displacement at dimensionless height \(\zeta=z/\)H at time t, EI is the flexural rigidity of the flexural beam, and the lateral stiffness ratio (\(\alpha\)) is a dimensionless parameter that determines the extent of flexure and shear deformations in the continuous model, thereby controlling the lateral deformation along the height. For \(\alpha=0\), the continuous model behaves as a Euler-Bernoulli beam, while \(\alpha\rightarrow\infty\) corresponds to a pure shear beam. Intermediate values of \(\alpha\) correspond to multistory buildings exhibiting combined flexure and shear deformations [5]. \[\alpha=H\sqrt{\frac{GA}{EI}} \tag{2}\]  
where GA is the shear rigidity of the shear beam. [PERSON] [4] specified the range of \(\alpha\) as 0 to 30 for conventional structures and defined different ranges on the basis of the structural system type. Shear wall systems exhibit \(\alpha\) values within \(0<\alpha<2\), while dual systems (moment-resisting frame-wall or moment-resisting frame-cross braced systems) fall within \(1.5<\alpha<6\). Moment-resisting frame systems typically have \(\alpha\) values ranging from 5 to 20, while \(\alpha\) values greater than 30 represent shear frame behavior. [PERSON] and [PERSON] [6,16] used the average values of these ranges according to the type of structural system in their building models. They suggested that this value can be used as 3.125 for dual systemsand 12.5 for moment-resisting frame systems. Since moment-resisting frame systems are analyzed in this study, [PERSON]'s \(\alpha\) approximation is considered as 12.5. For linear elastic structures, the responses can be obtained by combining the responses of all modes. The displacement expression (u(\(\zeta\),t)) at dimensionless height \(\zeta\) and time t for a continuous structure can be calculated using a linear combination of modal responses.