## 3 Estimation of Lateral Stiffness Ratio  
The displacement of buildings under horizontal loads is determined by the bending deformation of the columns and the rotation of the beams connected to the ends of the columns, which is influenced by their respective stiffness ratios. In a study conducted by [PERSON] [35], it was found that the structural and dynamic properties of framed buildings depend on the ratio of horizontal member (beam) stiffness to vertical member (column) stiffness. This ratio, known as the \"nodal rotation index\" or Blume coefficient (\(\rho\)), considers the structure's combined deformation.  
The beam-to-column stiffness ratio, \(\rho\), is a critical structural parameter that determines the relative rotation at the joint points in a building system [35]. It is expressed as the ratio of the total relative stiffness of the columns in the floor closest to the middle of the building height and the beams attached to the upper ends of these columns. The general equation for \(\rho\), assuming that beams and columns have the same modulus of elasticity, is given by Equation (18).  
\[\rho=\frac{\sum_{beams}\left(\frac{I_{\text{b}}}{I_{\text{b}}}\right)}{\sum_{ columns}\left(\frac{I_{\text{c}}}{I_{\text{c}}}\right)} \tag{18}\]  
where \(\text{I}_{\text{b}}\) and \(\text{I}_{\text{c}}\) represent the moment of inertia of the beam and column, respectively, and \(\text{I}_{\text{b}}\) and \(\text{h}_{\text{c}}\) represent the beam length and column height, respectively. The value of \(\rho\) indicates the relative stiffness of the beam-column system and determines the extent of participation of bending and shear deformations in the frame structure. A \(\rho\) value of zero indicates flexural deformation, a value of infinity (\(\infty\)) represents a shear structure, and intermediate values indicate a frame with combined shear and flexural deformation [4].  
The \(\rho\) parameter quantifies the relative stiffness between beams and columns, controlling the system's behavior and affecting the structure's natural period and mode shapes. The fundamental natural period provides valuable information about the dynamic properties of the frame system, including the proximity and distribution of natural periods and the shapes of natural modes. This parameter measures the relative beam-column stiffness and, thus, controls the degree of participation of lateral bending and shear deformations in a moment-resisting frame structure. In other words, it indicates how closely the system's behavior resembles that of a frame. The fundamental natural period of the system has a significant influence on the relative proximity or disparity of natural periods and shapes of natural modes, providing valuable information for the overall dynamic structural characteristics of the frame system [36; 37; 38; 39]. Therefore, \(\rho\) serves as a versatile variable for representing structural systems and retrieving important dynamic properties of the entire structure [39]. Unlike the dimensionless parameter \(\alpha\) used in continuous models, \(\rho\) is independent of the building height. In this study, the number of stories (N) is employed instead of height for framed buildings, and the estimation of \(\alpha\) is related to \(\rho\) and N.