## 2 Continuous Shear-Flexural Beam Model
### Implementation of Boundary Conditions  
When the continuous model is fixed at the base, the boundary conditions include displacement and rotation of zero at the base. However, in the proposed model, the displacement at the base is zero, while a rotation (\(\theta\)) is defined, which differs from zero. The following equations can express this:  
\[\phi_{i}(\zeta)_{\zeta=0}=0 \tag{12}\]  
\[\left.\frac{d\phi_{i}(\zeta)}{d\zeta}\right|_{\zeta=0}=\left.\frac{\phi_{i}(h_ {1}/H)}{h_{1}/H}\right. \tag{13}\]  
The case where the rotation is zero (\(\theta=0\)) provides the same solution as presented by [PERSON] and [PERSON] [5].  
Similarly, in the continuous model, at the free end, the boundary condition at the normalized height (\(\zeta=1\)) results in zero moment and shear force:  
\[\left.\frac{d^{2}\phi_{i}(\zeta)}{d\zeta^{2}}\right|_{\zeta=1}=0, \tag{14}\]  
\[\left.\left(\frac{d^{3}\phi_{i}(\zeta)}{d\zeta^{3}}-\alpha^{2}\frac{d\phi_{i} (\zeta)}{d\zeta_{\zeta}^{2}}\right)\right|_{\zeta=1}=0. \tag{15}\]  
The integration constants obtained using the boundary conditions are as follows:  
\[B_{2}=\left(-\frac{\beta_{i}\gamma_{i}^{2}\theta\sin(\gamma_{i})+\beta_{i}^{ 2}\gamma_{i}\theta\text{sinh}(\beta_{i})-2\beta_{i}^{2}\sin(\gamma_{i}\theta )\text{sinh}(\beta_{i})-2\gamma_{i}^{2}\sin(\gamma_{i})\text{sinh}(\beta_{i} \theta)}{\left(\gamma_{i}^{2}\cos(\gamma_{i})+\beta_{i}^{2}\cosh(\beta_{i} )\right)(\beta_{i}\theta-2\text{sinh}(\beta_{i}\theta))}\right)B_{1}\]  
\[B_{3}=\left(-\frac{\gamma_{i}\theta-2\sin(\gamma_{i}\theta)}{\beta_{i}\theta- 2\text{sinh}(\beta_{i}\theta)}\right)B_{1}\]  
\[B_{4}=-B_{2}\]  
The B\({}_{1-4}\) integration constants given in Equations (12)-(14) are used to substitute the mode shape function given in Equation (9) into the boundary condition Equation (15), resulting in the characteristic Equation (16).  
\[\left(-2\gamma_{i}\beta_{i}^{3}(\gamma_{i}\theta-\sin(\gamma_{i}\theta))+2 \gamma_{i}^{2}\beta_{i}^{2}\text{sinh}(\beta_{i}\theta)-\sin(\gamma_{i})\text {sinh}(\beta_{i})\left(\beta_{i}^{2}(\alpha^{2}\gamma_{i}\theta-2\beta_{i}^ {2}\sin(\gamma_{i}\theta))+2\gamma_{i}^{3}\beta\text{sinh}(\beta_{i}\theta) \right)+\right.\]  
\[\left.\cos(\gamma_{i})\cosh(\beta_{i})\left(\beta_{i}\left(-\left(\alpha^{4}+2 \gamma_{i}^{2}\beta_{i}^{2}\right)\theta\right)+2\gamma_{i}^{3}\sin(\gamma_{ i}\theta)\right)+2\beta_{i}^{4}\text{sinh}(\beta_{i}\theta)\right)=0 \tag{16}\]  
By solving the characteristic Equation (Equation (16)), which is a function of \(\alpha\) and \(\theta\), the eigenvalues (\(\gamma\)i) of the equation are obtained.  
The rotation (\(\theta\)) at the base is defined as the displacement ratio of the ground story. In this case, the expression \"\(\theta\)\" is defined as the ratio of the ground story height (h\({}_{1}\)) to the building height (H) through the normalized height (\(\zeta\)).  
\[\theta=\frac{h_{1}}{H}=\frac{h_{1}}{h_{1}+(N-1)h_{j}} \tag{17}\]  
When considering the case where all stories have equal story heights (h\({}_{1}=\)h\({}_{j}\)), the definition of \(\theta\) is equal to \"\(1/\text{N}\)\", where N is the number of stories. In this study, equal story heights are assumed for all stories.  
The continuous model with a uniform mass and stiffness distribution along the height, mode shapes, period ratios, and modal participation factors depends on dimensionless base rotation and lateral stiffness ratio parameters. The base rotation is defined as the drift ratio of the first story. The remaining lateral stiffness ratio needs to be known. In the literature, approximate ranges for this coefficient have been determined on the basis of the structural system of the building, but no definitive calculation has been suggested. In the subsequent parts of the study, a simple empirical equation is proposed for the lateral stiffness ratio using a curve fitting method to provide more accurate predictions of structural responses.