## 3 Estimation of Lateral Stiffness Ratio
### Determination of Lateral Stiffness Ratio (\(\alpha\))  
In the preceding section, the numerical values of lateral stiffness ratios, as defined in the proposed equations, were obtained via an iterative process employing SAP2000 simulations of building models with varying Blume ratios and floor quantities. Figure 4 presents the graph illustrating the variation of the lateral stiffness ratio (\(\alpha\)) for the number of floors (N) and the Blume ratio (\(\rho\)). It can be observed from the figure that \(\alpha\) changes depending on \(\rho\) and N. The relationship linking the lateral stiffness ratio (\(\alpha\)), the beam-column stiffness ratio (Blume coefficient) (\(\rho\)) specific to the building models, and the number of floors (N) is depicted in Figure 4.  
Figure 3: Comparison of maximum inter-story drift ratios (IDR\({}_{\text{max}}\)).  
Figure 2: (a) G5n5 sample model. (b) G9n8 sample model. (c) G9n20 sample model.  
The variation of \(\rho\), defined for discrete (frame) systems, significantly influenced the structural displacements. The lateral displacements increased as \(\rho\) increased, and the most significant differences in displacements occurred in the range of 0.0 to 0.125, where the fundamental mode transitioned from flexural to mixed (a combination of flexural and shear behavior). When \(\rho\) exceeded 0.125, the lateral displacements changed more gradually. Similar variations were also observed for the first mode of inter-story drift. The change in the maximum inter-story drift toward lower floors was rapid within the range of 0.0 to 0.125. In extreme cases, at \(\rho=0.0\), the structure behaved like a flexural beam, whereas, at \(\rho=\infty\), it exhibited pure shear behavior, with the maximum inter-story drift occurring at the top and ground floors, respectively [39]. These findings for moment-resisting frame systems support the relationship between the lateral stiffness ratio \(\alpha\) and \(\rho\). As shown in Figure 4, the variation of \(\alpha\), representing the structural behavior in the continuous model, was rapid for 0.0 \(<\)\(\rho<\) 0.125 and gradually increased after \(\rho\geq\) 0.125. Therefore, it is appropriate to consider the behavior in two sections to obtain a more accurate \(\alpha\).  
Nonlinear regression analysis using the DataFit [40] program was performed to estimate the equations. The relationship between the variables was investigated by defining two independent variables (\(\rho\) and N) and one dependent variable (\(\alpha\)). The most suitable set of equations to determine \(\alpha\) was obtained in two sections. The first Equation (21) corresponds to the range up flexural to the transition of mixed behavior, while the second Equation (22) corresponds to the range from mixed behavior to shear behavior.  
\[\alpha_{fc}(\rho,N)=a_{1}N^{b_{1}}\rho^{c_{1}}, \tag{21}\]  
\[\alpha_{c}(\rho,N)=a_{2}+b_{2}N+c_{2}\rho, \tag{22}\]  
When 0.0 \(<\)\(\rho<\) 0.125, a coefficient of multiple determination (R\({}^{2}\)) = 0.99 and adjusted coefficient of multiple determination (R\({}_{a}^{2}\)) = 0.99 were achieved. The corresponding equation constants were \"\(a_{1}\) = 2.672\", \"b\({}_{1}\) = 0.851\", and \"c\({}_{1}\) = 0.401\".  
When \(\rho\geq\) 0.125, a coefficient of multiple determination (R\({}^{2}\)) = 0.93 and adjusted coefficient of multiple determination (R\({}_{a}^{2}\)) = 0.93 were achieved. The corresponding equation constants were \"\(a_{2}\) = 0.0946\", \"b\({}_{2}\) = 0.877\", and \"c\({}_{2}\) = 2.722\".  
With this proposed estimation equation, the lateral stiffness ratio of a building model can be quickly determined with minimal information from the building model. It provides more accurate results than the approximate average values commonly used in the literature and assists in estimating \(\alpha\) without involving complex calculations for the shear stiffness expression of frame systems (GA).  
Figure 4: Relationship linking \(\alpha_{\text{est}}\), N, and \(\rho\) for sample building models.