## 3 Estimation of Lateral Stiffness Ratio
### Prediction of Structure Responses  
To evaluate the predictive capability of the proposed model for structural responses, we obtained solutions for various sets of sample buildings, as created in the preceding section. The SAP2000 software was employed to analyze these models, encompassing a range of 5-20 stories, as specified in Table 1. Subsequently, the results obtained from these analyses were utilized in this study to make predictions using equations formulated on the basis of the proposed base rotation (\(\theta\) = 1/N) and lateral stiffness ratio (\(\alpha\)). The optimal lateral stiffness ratio was determined through an iterative trial-and-error approach to ensure an accurate prediction of structural behavior. The primary aim was to identify the lateral stiffness ratio that yields the most accurate prediction of the structural response. Primarily, by comparing the mode shape profiles derived from the SAP2000 (FEM) models with the mode shape functions of the proposed model, it is estimated by the \(\alpha\) values through iterative experimentation to achieve the closest correspondence between the results. \begin{table}
\begin{tabular}{c c} \hline
**Model** & \(\rho\) \\ \hline
0. Group (G0) & 0.004 \\
1. Group (G1) & 0.03 \\
2. Group (G2) & 0.06 \\
3. Group (G3) & 0.13 \\
4. Group (G4) & 0.30 \\
5. Group (G5) & 0.53 \\
6. Group (G6) & 0.76 \\
7. Group (G7) & 1.00 \\
8. Group (GS) & 1.50 \\
9. Group (G9) & 2.07 \\ \hline \end{tabular} Note: Within the study context, the models were named on the basis of the group number and the value of N. For example, the N = 10 story model in Group 7 is denoted as “G7n10”, while the N = 5 story model in Group 5 is encoded as “G5n5”.  
\end{table} 
Table 1: Generic building model information. Figure 1: UBC97 response spectrum. In this study, dynamic analysis employing the response spectrum method was performed to obtain the first-mode (i = 1) lateral displacement (u\({}_{i}\)(\(\zeta\),T)) and inter-story drift ratio (IDR\({}_{i}\)(\(\zeta\),T)), as defined in Equations (19) and (20), respectively. \[u_{i}(\zeta,T_{i})=\Gamma_{i}\phi_{i}(\zeta)S_{d,i}(T_{i}) \tag{19}\]  
\[IDR_{i}(\zeta,T_{i})=\Gamma_{i}S_{d,i}(T_{i})\left(\frac{\phi_{i}(\zeta_{j})- \phi_{i}(\zeta_{j-1})}{\zeta_{j}-\zeta_{j-1}}\right) \tag{20}\]  
We obtained closed-form solutions for the continuous flexural-shear beam utilizing the MATHEMATICA software. The lateral displacement, inter-story drift ratio, and mode shapes were determined and visually presented. Furthermore, the SAP2000 structural analysis program, which utilizes finite element-based solutions, was employed to dynamically analyze the structural models following the UBC97 response spectrum. The resultant structural responses were considered \"exact\" and compared with the solutions obtained from the continuous model. In the lateral displacement expression given by Equation (19), if S\({}_{d}\)(T\({}_{i}\)) is known in the model with a constant mass distribution along the height, the behavior of the structure depends only on \(\alpha\)[4; 5]. We compared the lateral displacement profile of the SAP2000 model (u(z,T)/u(H,T)) with the mode shape profile of the continuous model (\(\phi\)) and the drift mode shape (u(z\({}_{i}\),T) - u(z\({}_{j-1}\),T)/h\({}_{j}\)) (\(\phi^{\prime}\)). The \(\alpha\) values employed in the continuous model were estimated through iterative experimentation. Similarly, when the modal displacement profile of the SAP2000 model was divided by the spectral displacement (u(z,T)/S\({}_{d}\)(T)), it was compared with the modal displacement of the continuous model (\(\Gamma\phi\)), while the modal drift ratio profile (u(z\({}_{i}\),T) - u(z\({}_{j-1}\),T)S\({}_{d}\)(T) \(\times\) /h\({}_{j}\)) (\(\Gamma\phi^{\prime}\)) was compared with the drift mode shape of the continuous model.