# Damping and Natural Period Evaluation of Tall RC Buildings Using Full-Scale Data in Korea
## 4 Natural Period and Damping Ratio Identification from Measured Full-Scale Data
### Equation for Damping Ratio Estimation  
Figure 10 presents the results of a regression analysis of 36 measured damping ratios and 15 damping ratios obtained from references [4,5]. Similar to the natural period regression, the linear regression analysis of damping ratio was performed such that the damping value was linearly proportional to the inverse of building height, which is linearly proportional to the natural frequency of the building, as has been considered in previous research [5,13,14]. Figure 10 shows that the damping ratio decreases as the building height increases. The proposed equation of damping ratio for the wind design of RC buildings obtained from the regression analysis is as follows:  
\[\zeta=0.2467/H+0.0067 \tag{21}\]  
Figure 10: Regression analysis of measured damping ratio.  
Figure 9: Comparison of proposed equation for natural frequency estimation with other empirical formulae.  
The coefficient of correlation of Equation (21) is 0.6461. Even though the coefficient of correlation is relatively low compared to that in Equation (19) for natural period estimation, the value is considerably greater than that yielded by the Japanese formula of [PERSON] et al. Equation (21) can be rewritten considering the relationship between natural period and building height in Equation (20) as follows:  
\[\zeta=0.00487f_{1}+0.0067 \tag{22}\]  
Notably, Equation (22) does not include a response-dependent term unlike the empirical formulae in Equations. (10) and (16) of [PERSON] et al. and [PERSON], respectively. Figures 11 and 12 compare the proposed Equation (21) with those of [PERSON], [PERSON] et al., and [PERSON]. Only the frequency-dependent term is considered in Figure 11, while the response-dependent term is included in Figure 12 in the empirical formulae of [PERSON] et al. and [PERSON]. The value of x/H = \(2\times 10^{-5}\), as suggested by [PERSON] et al., is used for tip displacement. In addition, the empirical formula proposed by [PERSON] is compared in Figure 11.  
Figure 11 shows that the empirical formula of [PERSON] without the amplitude-dependent term is the closest to the equation proposed in this study. [PERSON]'s formula, however, overestimates the damping ratio when the amplitude-dependent term is included in the damping ratio estimation, as shown in Figure 11. Moreover, the empirical formula of [PERSON] yields very large damping ratios for buildings taller than 100 m because of the Rayleigh damping consideration. Note that only the first modal frequency calculated from Equation (5) is used to evaluate the empirical formula of [PERSON] in Equation (8).  
If the amplitude-dependent term is included, as in Figure 12, the empirical formula provided in ESDU 83009 matches well with the proposed equation. The empirical formulae provided in AIJ 2000 and Yoon considerably overestimate the damping ratio of buildings taller than 200 m. This overestimation can mainly be ascribed to the fact that those formulae were derived using the data of buildings with heights ranging up to 66 m for Yoon and up to 129.8 m for AIJ 2000.  
Figure 11: Comparison of proposed equation for estimating damping ratio with various empirical formulae—only the frequency-dependent term is included.