An innovative analytical approach for predicting the fundamental time period of moment-resisting frames

The fundamental period of vibration of a building is one of the important parameters appearing in the different building codes to compute the design base shear and lateral loads. The empirical formulae found in codes are mostly based on the building type (e.g., frame or shear wall), overall size, and construction material (e.g., steel or reinforced concrete). The determination of this period is critical since it has a direct impact on the calculation of seismic loads for improving both the accuracy and reliability of the building design process.
Almost all the formulas currently found in the "Uniform Building Code" (UBC) [1], the "Structural Engineers Association of California" (SEAOC) [2] recommendations, and the "Egyptian Code" (EGC) [3] are primarily derived from building vibration data recorded during the 1971 San Fernando earthquake. Later, in 1996 and 1997, Goel and Chopra [4, 5] evaluated existing empirical formulas and subsequently proposed new empirical formulas to estimate the fundamental vibration period of reinforced concrete moment-resisting frame (RC MRF) buildings. These methods were developed for use in equivalent lateral force analysis, as required by building codes, utilizing motion data from various structures recorded during multiple earthquakes. The data analyzed by [4] were obtained from earthquakes, including the 1971 San Fernando, 1984 Morgan Hill, 1986 Mt. Lewis and Palm Springs, 1987 Whittier, 1989 Loma Prieta, 1990 Upland, 1991 Sierra Madre, and 1994 Northridge events.
Salama [6] also updated formulas for the estimation of the fundamental vibration period by using regression analysis on data from eight recorded earthquakes in California. Their formulas account for both the building height and the number of stories in the moment-resisting reinforced concrete frames. Several other researchers have also proposed an adjustment to the Ct coefficient based on the number of stories, according to the formulas in American and Egyptian building codes. Similarly, Kalpan et al. [7] emphasized that the empirical formulas in building codes should be based on the particular region in which the typical design and architectural character of local construction are exhibited. In this regard, they derived a simplified formula to estimate the fundamental elastic period of mid-rise reinforced concrete buildings and compared it with the formula available in the "Turkish Earthquake Code" (TEC) [8]. Their study indicated that the empirical formula in the code yields non-conservative base shear values. They also investigated the contribution of infill walls to the lateral stiffness of buildings and offered preliminary recommendations based on their findings.
Several researchers, including $[9,10,11,12,13,14,15,16]$$[9,10,11,12,13,14,15,16]$[9,10,11,12,13,14,15,16][9,10,11,12,13,14,15,16]$[9,10,11,12,13,14,15,16]$, have derived simplified equations based on building height by using ambient vibration measurements to estimate the fundamental period of buildings. Mohamed et al. [17] conducted nonlinear dynamic analyses to estimate the fundamental period of mid-rise moment-resisting RC frames. In their study, the results were compared with the period equations given by Salama and "American Society of Civil Engineers: Minimum Design Loads for Buildings and Other Structures" (ASCE 7-16) [6, 18]. They developed several formulas from the regression analysis of their test results to estimate the period of vibration. They also supported the application of rational approaches,
like modal analysis, to calculate the fundamental period of RC frames and, in this way, remove some of the restrictive codebased limits.
Young and H. Adeli [19, 20] criticized the methodology for determining the estimated fundamental period of momentresisting frames obtained from ASCE 7-16 [18] as being overly conservative. They emphasized that this approach doesn't fully account for geometric irregularity in structures, which may greatly affect the structural responses during seismic events. Due to this deficiency, an extensive study was carried out on the fundamental period of moment-resisting frames with various types of geometric irregularities. They developed several formulas that took into consideration both vertical and horizontal irregularities; hence, these formulas became more accurate and flexible methods of estimating the fundamental period.
The degree of irregularity in a stepped building frame is estimated in [21], using the Rayleigh approach , and suggested modifications to the empirical formula used in building codes for estimating the fundamental period of such stepped frames. However, the calculation of the regularity index, as proposed by them, is complex and time-consuming. Ricci et al. [22] studied the elastic vibration period of uncracked and infilled reinforced concrete moment-resisting frame buildings. They employed a modal analysis approach to assess the structural response and subsequently offered simplified formulas to estimate the fundamental period. These formulas were based on regression analysis of numerical data, providing a practical alternative for efficient computation.
An equation was proposed for estimating the fundamental vibration period of regular frame structures with a maximum of ten stories, aiming to improve the empirical expressions widely used in building codes [23]. They utilized structural features such as stiffness and mass distribution, which enable more accurate estimations to be obtained by getting a correction factor based on effective mass and height, and the effective lateral stiffness of the structure. They also saw its applicability to fairly irregular frame structures with minimal loss of accuracy.
Alrudaini [24] critically examined the estimation of the fundamental vibration period ( T ) of reinforced concrete momentresisting frame (RC-MRF) buildings, noting the limitation in existing seismic design codes. Based on regression analysis of recorded motions from RC moment-resisting frame (RC-MRF) buildings during eight California earthquakes, an improved formula was proposed in [24] that implicitly accounts for the floor height (h) as a significant parameter by incorporating both the total building height $(\mathrm{H})$$(\mathrm{H})$(H)(\mathrm{H})$(\mathrm{H})$ and the number of stories $(\mathrm{N})$$(\mathrm{N})$(N)(\mathrm{N})$(\mathrm{N})$. The study suggests the modification of the coefficient Ct in the current code equations as a function of N to enhance the precision of T estimates.
Current studies have focused more on improving the precision of fundamental period estimation of steel and reinforced concrete structures, particularly in terms of complex structural properties such as height, irregularity, and bracing systems. Kumar et al. [25] performed a comparative analysis of machine learning algorithms, including artificial neural networks (ANN), genetic programming (GP), and regression trees (RT), demonstrating that ANN models exhibited superior predictive accuracy, especially in the presence of geometric irregularities.
Likewise, Rahman et al. [26] employed interpretable machine learning methods with SHapley Additive exPlanations (SHAP) to determine the significant structural parameters influencing the fundamental period in steel-braced reinforced concrete buildings and the way its configurations have significant influences on its dynamic behavior. In order to accurately capture the uncertainties associated with aging and material degradation, Shan et al. [27] developed probabilistic machine learning techniques to forecast the natural period of existing high-rise reinforced concrete buildings. Moreover, Karampinis et al. [28] created hybrid analytical-machine learning models that used SHAP values to generate interpretable equations for framed structures. This allowed them to preserve engineering transparency while achieving prediction accuracy on par with purely data-driven methods. These developments highlight a growing movement to improve period estimation, especially for complex and irregular structures, by combining data-driven techniques with conventional analytical models.
Most seismic design codes recommend empirical equations for the estimation of the fundamental vibration period of buildings. However, it has been demonstrated through several studies that these code-based equations typically underestimate significantly the actual measured periods, as stipulated by Goel and Chopra [4]. They once again highlighted the inadequacy of the code provisions available by citing that the simple equations that are commonly used can vary as much as $100\mathrm{\%}$$100\mathrm{\%}$100%100 \%$100\mathrm{\%}$ from observed values. These broad variations can lead to excessively conservative estimates of base shear force, thus structurally over-strengthened and uneconomical designs. Although several equations have been presented both in design codes and in the literature, many are empirical and usually derived from regression analyses of small datasets. These formulations often neglect the physical parameters most critical to dynamic behavior. Although many recent studies have also looked into data-driven and machine learning strategies can provide high predictive accuracy, their applicability and reliability in engineering practice are limited by the need for large, high-quality datasets and the possibility of overfitting or a lack of physical interpretability. Furthermore, they can be computationally intensive and require specialized knowledge to implement, which makes them less useful for everyday design. Hence, there is an evident necessity for a more logical and
physically based model, including the equivalent mass and stiffness, two of the most significant parameters that control the fundamental period, precisely to increase both accuracy and usability of period estimation techniques.
This paper proposes a novel analysis procedure for the accurate estimation of the fundamental period of vibration of reinforced concrete moment-resisting frames. The equivalent seismic mass and the building's lateral stiffness are both included in the proposed procedure for higher accuracy than the current procedure. This approach differs from the previous approaches because it provides an easier process to compute the equivalent stiffness and equivalent mass of a multi-story building and also computes its fundamental period with accuracy, which gives more accurate results, and the resulting equations are in the form of closed-form expressions. Sensitivity analysis was carried out to investigate the influence of significant design parameters such as building height, span length, beam-to-column stiffness, and material elasticity. This procedure not only serves to confirm the strength and validity of the proposed equation but also establishes the key parameters that need to be taken into consideration in the design. It confirms the proposed model as being stable and representative of a wide range of buildings, thus making it more generalizable and usable in practice.

The empirical formulas for the fundamental vibration period of moment-resisting frames (MRFs), adopted by most design codes, including U.S. building codes, UBC [1], the "Applied Technology Council" (ATC,) [29], SEAOC, [2], the "National Earthquake Hazards Reduction Program" (NEHRP) [30], and the more recent EGC [3], are generally in the form:

In this empirical formula, H represents the total height of the building in meters, which is measured as the vertical distance from the ground level or foundation. Ct represents a numerical coefficient related to the lateral force-resisting system. Values adopted for Ct by the above codes are 0.047 for RC MRF and 0.051 for Steel MRF buildings. For some design codes, such as the NEHRP [30] regulations and earlier versions of other seismic codes, including the EGC [3], an alternative formula is available for RC MRF buildings. This formula expresses the fundamental period as:

The simple formula is restricted to structures with a height limit of a story of 10 feet and a height of not more than 12 stories, where N denotes the number of stories. Salama [6] provides the estimation of the fundamental period of vibration (T) of reinforced concrete moment-resisting frame buildings, one of the more important parameters in seismic design. Conventional seismic codes, such as ASCE [18] and EGC [3], estimate T using empirical equations that account for either building height ( H ) or the number of stories ( N ) individually, not both together. An improved empirical formula that includes both H and N , based on regression analysis of the data for moment-resisting frame reinforced concrete buildings experiencing eight California earthquakes was suggested in [6]. The method is intended to give an improved estimate of T by considering variations in floor heights and the number of stories. The study recommends that the coefficient Ct of the current code formulas be modified for the effect of N in order to make the correlation between estimated and observed vibration periods even better.
Salama [6] adopted a theoretical form of the period equation that included both the number of stories ( N ) and the total height $(\mathrm{H})$$(\mathrm{H})$(H)(\mathrm{H})$(\mathrm{H})$ of the building

in this approach, the constants $\mathrm{a},\mathrm{b}$$\mathrm{a},\mathrm{b}$a,b\mathrm{a}, \mathrm{b}$\mathrm{a},\mathrm{b}$, and c as variables in an unconstrained regression analysis, allowing the optimal fit of statistical data to measured data. This method minimized the standard error of the estimate, resulting in:

where T is the fundamental period of vibration (in seconds), N is the number of stories, and H is the height of the building (in meters). Aninthaneni et al.[23] derived an improved analytical method to forecast the fundamental period (T) of regular
reinforced concrete (RC) moment-resisting frame buildings in the Bulletin of the New Zealand Society for Earthquake Engineering. The recommended equation estimates T more correctly than empirical equations by considering seismic weight, storey height, number of storeys, bays, and effective stiffness. The method is validated against eigenvalue analysis and experimental findings on low- to mid-rise RC frames, even with minor deviation.
The improved empirical formulas proposed are:

In this equation, ${\varphi }_3$${\varphi }_3$phi_(3)\phi_{3}${\varphi }_3$ is a correcting factor that takes into account the effective distribution of mass and height in the structure. The variable Ws is the total seismic weight of the frame, and $h$$h$hh$h$ is the storey height, or the height between two consecutive floors. The storey number is denoted by ${\mathrm{n}}_{\mathrm{s}}$${\mathrm{n}}_{\mathrm{s}}$n_(s)\mathrm{n}_{\mathrm{s}}${\mathrm{n}}_{\mathrm{s}}$, meanwhile, ${\mathrm{n}}_{\mathrm{b}}$${\mathrm{n}}_{\mathrm{b}}$n_(b)\mathrm{n}_{\mathrm{b}}${\mathrm{n}}_{\mathrm{b}}$ is the total number of bays in the frame. The parameters ${\beta }_{\mathrm{d}}$${\beta }_{\mathrm{d}}$beta_(d)\beta_{\mathrm{d}}${\beta }_{\mathrm{d}}$ and ${\beta }_c$${\beta }_c$beta_(c)\beta_{c}${\beta }_c$ are the depiction of the distribution of stiffness over the building height for the beam and column systems, respectively. The ratio of the beam-to-column stiffness is given by $\lambda$$\lambda$lambda\lambda$\lambda$, and the material properties are described by ${\mathrm{E}}_{\mathrm{c}}$${\mathrm{E}}_{\mathrm{c}}$E_(c)\mathrm{E}_{\mathrm{c}}${\mathrm{E}}_{\mathrm{c}}$, the modulus of elasticity of the concrete, and ${\mathrm{I}}_{\mathrm{cef}}$${\mathrm{I}}_{\mathrm{cef}}$I_(cef)\mathrm{I}_{\mathrm{cef}}${\mathrm{I}}_{\mathrm{cef}}$, the effective moment of inertia of the columns. The model enables the prediction of the fundamental period of the building, considering both material and geometric properties, with better accuracy compared to standard empirical approaches for common moment-resisting frame-type structures.
Goel and Chopra [4] experimentally validated the equations provided in U.S. seismic codes for approximating the fundamental period of vibration (T) of buildings. Based on the assessment of observed periods of 106 buildings that experienced eight California earthquakes from 1971 through 1994, they discovered that the current code equations underestimate the fundamental time period. By regression analysis of this expanded database, [5] developed better empirical equations that more accurately estimate the natural periods of reinforced concrete (RC) and steel moment-resisting frame (MRF) buildings. These improved equations had a significant impact in revising seismic design provisions in, for example, the NEHRP [31] and ASCE 7-05 [32] standards, and are expressed as:

Here, T is the fundamental period of vibration of the building in seconds, and H is the height of the building in meters. Most current design codes specify that the design base shear should be calculated using the following equation:

where V is the design base shear, W is the total seismic dead load, and C is the seismic coefficient. The seismic coefficient C depends on several parameters: the soil profile, the seismic zone factor, the importance factor, the building's fundamental period T , and a numerical coefficient representing overstrength and global ductility capacity inherent in the lateral loadresisting system. The fundamental period Ta calculated by the empirical formulas from equations (1) and (2) is typically designed to be shorter than the true period in order to provide a conservative estimate of the base shear, as structures are required to be designed for larger seismic forces. Thus, code formulas are intentionally calibrated to underestimate the fundamental period by about 10-20%. Furthermore, while building codes allow the determination of the fundamental period using rational methods, such as Rayleigh's method [33], they also specify that the period computed based on these methods shall not exceed the period obtained from the empirical formula by more than a certain factor. In this way, consistency in seismic design is achieved, while there is provision for making more accurate calculations when required.

To evaluate the validity of frequently used empirical formulas in identifying the fundamental period of structures, comparative study was performed between code expressions and FEM. The comparison given in Figure 1 shows a detailed assessment of accuracy of different empirical expressions to comprehensive finite element method (FEM) analysis for calculation of the fundamental period of structures for a broad range of heights. Specifically, the comparison incorporates empirical period formulae from UBC [1], ASCE 7-05 [32], and also Salama's equation [6] as written in Equations (1), (5) and (3) respectively. Figure 1 presents several important observations concerning the accuracy of these formulae relative to FEM results:

Seismic design typically begins with estimating a trustworthy seismic force, usually based on the periods of the building's vibration. These periods are primarily influenced by the mass and stiffness of the building. In this case, seismic design often becomes a process of trial and error, since the relationship between seismic capacity, mass, and stiffness must be well recognized to ensure effective seismic performance. To avoid unnecessary trial and error, most standards include starting points for element size, construction orientation, number of levels, and material properties. These provide natural period calculations, and many formulas from codes and literature are used to calculate the building's seismic load. The fundamental period of the building, during which a significant portion of the mass participates in the associated mode of vibration, is by definition the natural period of vibration. While more complex formulas are available in building codes, simpler equations based on height or the number of stories are often recommended to streamline the calculation process. Several studies have examined the effect of both structural and non-structural factors on the fundamental period of vibration in reinforced concrete buildings. Consequently, the empirical equations have revealed that the natural period is mostly determined by the building's height or number of stories. More specifically, the taller or higher a building is, the longer the period of vibration will be [34, 35]. These two properties, height and number of stories, are found to be the two most influential ones that determine a building's fundamental period. Consequently, most design codes rely on one of these two properties, preferably height or number of floors, in developing their empirical equations for the vibration period. Notable examples of such codes include the "building standard law of Japan" (BSLJ) [36], the National Building Code of Canada (NBCC, 1995) [37] the Uniform UBC, [1], and the ATC [29], all of which incorporate height or number of stories as key parameters in their period formulas.

The most practical approach to developing an analytical model of a real-frame structure is to model it as a skeleton with masses lumped at each floor level. In this section, a simplified technique is outlined for calculating the fundamental period of vibration of such structures. As previously mentioned, and as widely recognized, the period of vibration of a building depends on its mass and stiffness. For an idealized Single-Degree-of-Freedom (SDOF) system, the fundamental period is classically expressed by Equation (7), which defines the dynamic relationship between mass and stiffness.

Equation (7) forms the base of this study. An RC building can be modeled by a single-degree-of-freedom (SDOF) system using an equivalent lumped mass ${\mathrm{m}}_{\text{eq}}$${\mathrm{m}}_{\text{eq }}$m_("eq ")\mathrm{m}_{\text {eq }}${\mathrm{m}}_{\text{eq }}$ and equivalent stiffness ${\mathrm{k}}_{\text{eq}}$${\mathrm{k}}_{\text{eq }}$k_("eq ")\mathrm{k}_{\text {eq }}${\mathrm{k}}_{\text{eq }}$ as shown in Fig. 2. This idealization is concerned with the fundamental period of vibration and the interaction between the mass and stiffness in the system, making it much easier to analyze the dynamic behavior of the building.

Procedure for estimating the equivalent lateral stiffness of the frame ( $K_{\text{eq}}$$K_{\text{eq }}$K_("eq ")K_{\text {eq }}$K_{\text{eq }}$ )
The following explanation may help clarify the development of the current formula. Initially, the building is simplified to achieve a more accurate estimate of lateral stiffness. The proposed method begins by calculating the equivalent stiffness for a two-storey building, as shown in Fig. 3. Using the Rayleigh approach, the process for estimating the fundamental period of the building is formulated in this section.

To develop the shear stiffness matrix of the two-storey MRF, considering the stiffness of the beams and anti-symmetry, the flexibility matrix is developed first by applying two horizontal forces separately and calculating the horizontal deformations. The stiffness matrix of the frame due to translational and rotational degrees of freedom takes the form:

To obtain the flexibility matrix, the horizontal deformations due to horizontal unit forces are obtained as follows:

where T is the transformation matrix that takes the form:

Then the shear stiffness matrix of the shown frame is obtained as follows:

The structure mass is condensed to one mass at the first-floor level, then the structure mass matrix takes the form:

where $m_1$$m_1$m_(1)m_{1}$m_1$ is the equivalent mass of the structure.
Then the first natural frequency of this frame is obtained by:

Then the equivalent first-floor stiffness is obtained as follows:

which takes the following form:

where ${\mathrm{K}}_{\mathrm{eq}}$${\mathrm{K}}_{\mathrm{eq}}$K_(eq)\mathrm{K}_{\mathrm{eq}}${\mathrm{K}}_{\mathrm{eq}}$ is the lateral equivalent stiffness, E is the concrete modulus of elasticity, ${\mathrm{Ic}}_1$${\mathrm{Ic}}_1$Ic_(1)\mathrm{Ic}_{1}${\mathrm{Ic}}_1$ and ${\mathrm{Ic}}_2$${\mathrm{Ic}}_2$Ic_(2)\mathrm{Ic}_{2}${\mathrm{Ic}}_2$ are the moments of inertia of the first and second floors columns, respectively, L is the frame span, ${\mathrm{Ib}}_1$${\mathrm{Ib}}_1$Ib_(1)\mathrm{Ib}_{1}${\mathrm{Ib}}_1$ and ${\mathrm{Ib}}_2$${\mathrm{Ib}}_2$Ib_(2)\mathrm{Ib}_{2}${\mathrm{Ib}}_2$ are the moments of inertia of the first and second floors girders, respectively, and ${\mathrm{h}}_1$${\mathrm{h}}_1$h_(1)\mathrm{h}_{1}${\mathrm{h}}_1$ and ${\mathrm{h}}_2$${\mathrm{h}}_2$h_(2)\mathrm{h}_{2}${\mathrm{h}}_2$ are the heights of the first and second floors, respectively. By substituting in equation (15) with ${\mathrm{Ic}}_1={\mathrm{Ic}}_2$${\mathrm{Ic}}_1={\mathrm{Ic}}_2$Ic_(1)=Ic_(2)\mathrm{Ic}_{1}=\mathrm{Ic}_{2}${\mathrm{Ic}}_1={\mathrm{Ic}}_2$ and ${\mathrm{Ib}}_1={\mathrm{Ib}}_2$${\mathrm{Ib}}_1={\mathrm{Ib}}_2$Ib_(1)=Ib_(2)\mathrm{Ib}_{1}=\mathrm{Ib}_{2}${\mathrm{Ib}}_1={\mathrm{Ib}}_2$ to obtain a more simplified form as follows:

By substituting into equation 16 with ${\mathrm{I}}_{\mathrm{b}1}={\alpha }^{\ast }{\mathrm{Ic}}_1$${\mathrm{I}}_{\mathrm{b}1}={\alpha }^{\ast }{\mathrm{Ic}}_1$I_(b1)=alpha^(**)Ic_(1)\mathrm{I}_{\mathrm{b} 1}=\alpha^{*} \mathrm{Ic}_{1}${\mathrm{I}}_{\mathrm{b}1}={\alpha }^{\ast }{\mathrm{Ic}}_1$.

where $\alpha$$\alpha$alpha\alpha$\alpha$ is defined as the beam-to-column moment of inertia ratio.
This method is applied on a broader scale to include multi-story frames, as shown in Fig. 4. The equivalent lateral stiffness is derived using the same equation, based on the sections and heights of the first and second floors. This indicates that the
upper floors have a negligible impact on determining the lateral stiffness, allowing for a simplified analysis focused primarily on the lower floors. This assumption helps streamline calculations while maintaining reasonable accuracy for structural behavior under lateral loads. It becomes clear that the previous equation for ${\mathrm{K}}_{\mathrm{eq}}$${\mathrm{K}}_{\mathrm{eq}}$K_(eq)\mathrm{K}_{\mathrm{eq}}${\mathrm{K}}_{\mathrm{eq}}$, expressed in terms of the properties and behaviors of the first two stories, can be generalized for multi-story structures with high accuracy.

Equivalent seismic mass is a simplified representation of how each floor's weight contributes to the building's overall seismic response. It considers the distribution of mass and how it affects the structure's dynamic behavior during an earthquake. The sequence of the method is as illustrated in Fig. 5, where the following building consists of three floors. Two iterations are performed to obtain an equivalent mass concentrated at the level of the first floor. The first iteration involves shifting the mass of the third floor to the level of the second floor to obtain ${\mathrm{m}}_{\mathrm{eq}}$${\mathrm{m}}_{\mathrm{eq}}$m_(eq)\mathrm{m}_{\mathrm{eq}}${\mathrm{m}}_{\mathrm{eq}}$. Then, the mass obtained from the previous iteration, ${\mathrm{m}}_{\text{eq}1}$${\mathrm{m}}_{\text{eq }1}$m_("eq "1)\mathrm{m}_{\text {eq } 1}${\mathrm{m}}_{\text{eq }1}$, is shifted to the level of the first floor to calculate ${\mathrm{m}}_{\text{eq}2}$${\mathrm{m}}_{\text{eq }2}$m_("eq "2)\mathrm{m}_{\text {eq } 2}${\mathrm{m}}_{\text{eq }2}$, resulting in a single equivalent mass at the first-floor level.

To perform this task, return again to the two-storey frame shown in Fig. 3, but the mass of the second floor is set by ${\mathrm{m}}_2$${\mathrm{m}}_2$m_(2)\mathrm{m}_{2}${\mathrm{m}}_2$, and the mass of the first floor is omitted; then the structure mass matrix takes the form:

Again, by substituting into the eigen equation:

Then the equivalent mass is expressed as:

To get the mass contribution factor, apply the following equation:

By substituting into Eqn. (21) for $\alpha ={\mathrm{I}}_{\mathrm{b}}/{\mathrm{I}}_{\mathrm{c}}$$\alpha ={\mathrm{I}}_{\mathrm{b}}/{\mathrm{I}}_{\mathrm{c}}$alpha=I_(b)//I_(c)\alpha=\mathrm{I}_{\mathrm{b}} / \mathrm{I}_{\mathrm{c}}$\alpha ={\mathrm{I}}_{\mathrm{b}}/{\mathrm{I}}_{\mathrm{c}}$ and ${\beta }_2=\mathrm{L}/{\mathrm{h}}_2$${\beta }_2=\mathrm{L}/{\mathrm{h}}_2$beta_(2)=L//h_(2)\beta_{2}=\mathrm{L} / \mathrm{h}_{2}${\beta }_2=\mathrm{L}/{\mathrm{h}}_2$, obtaining:

To simplify calculations, the formula can be modified using ${\mathrm{h}}_1=\mathrm{n}\cdot {\mathrm{h}}_2$${\mathrm{h}}_1=\mathrm{n}\cdot {\mathrm{h}}_2$h_(1)=n*h_(2)\mathrm{h}_{1}=\mathrm{n} \cdot \mathrm{h}_{2}${\mathrm{h}}_1=\mathrm{n}\cdot {\mathrm{h}}_2$ in Eqn. (22), where ${\mathrm{h}}_2$${\mathrm{h}}_2$h_(2)\mathrm{h}_{2}${\mathrm{h}}_2$ is the floor height, ${\mathrm{h}}_1$${\mathrm{h}}_1$h_(1)\mathrm{h}_{1}${\mathrm{h}}_1$ is the height at which the mass has accumulated from the previous floor, and $n$$n$nn$n$ is the floor number to which the mass was moved to simplify the calculation.

The following equation is obtained by replacing $\gamma =\frac{{\beta }_2}{\alpha }$$\gamma =\frac{{\beta }_2}{\alpha }$gamma=(beta_(2))/(alpha)\gamma=\frac{\beta_{2}}{\alpha}$\gamma =\frac{{\beta }_2}{\alpha }$, in Eqn. (23):

A simplified form of Equation (22) can be obtained by setting the stiffness ratio $\alpha =1$$\alpha =1$alpha=1\alpha=1$\alpha =1$, which corresponds to the case where the beam and column sections are identical. This assumption leads to the following simplified expression:

By substituting in Eq. (24) with ${\mathrm{h}}_1={\mathrm{n}}^{\ast }{\mathrm{h}}_2$${\mathrm{h}}_1={\mathrm{n}}^{\ast }{\mathrm{h}}_2$h_(1)=n^(**)h_(2)\mathrm{h}_{1}=\mathrm{n}^{*} \mathrm{~h}_{2}${\mathrm{h}}_1={\mathrm{n}}^{\ast }{\mathrm{h}}_2$