# 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
An output-only system identification technique called stochastic subspace identification (SSI) [20] was used to identify the natural period and damping ratio. Because wind load is generally assumed to be a wideband random signal with an averaging time of 10 min, the ambient vibration of tall buildings subjected to wind load can be applied to output-only system identification methods, such as the SSI method [21]. The SSI method yields estimates of modal parameters obtained from singular value decomposition of the block Hankel matrix, which is composed of correlation matrices of responses. Figure 4: Measured acceleration time histories of Central Park 103: (**a**) x-direction and (**b**) y-direction. Figure 3: Measured acceleration time histories of Harbor View 1502: (**a**) x-direction and (**b**) y-direction. Because the dynamic parameters are extracted from the state-space equation obtained directly from the time series of the data, this method is often used for system identification when the input force is not known [22]. Stabilization charts shown in Figure 5 were created to classify the poles into estimated poles by repeating the identification analysis according to the ascending order of state-space equations used in the SSI method. Estimated poles that correspond to physically relevant modes tend to appear for each order of state-space equations at nearly identical frequencies, while the so-called mathematical poles, which result from mathematical solutions of the equation and are meaningless with respect to the physical interpretation, tend to jump around [21]. Note that the first two modes of Central Park 103 are well separated while those of Harbor View 1502 are closely distributed as shown in Figure 5. The close distribution of modes is common in tall buildings with square plans. The identified first mode natural periods and damping ratios of the measured buildings are listed in Table 6.  
In addition to the results of the 36 buildings presented in Table 7, the data of 22 buildings used in [5,6] were added to include more data on buildings shorter than 100 m. The natural periods of all 22 buildings were provided in [5], where a formula to compute the natural period for wind design was proposed, while the damping ratios of 15 of those 22 buildings were given in [6], where a formula to compute the damping ratio for wind design was proposed. Consequently, the data of 51 buildings for evaluation of the damping ratio formulae and those of 58 buildings for evaluation of the natural period were used in this paper. The heights of the 58 buildings ranged from 24.4 m to 305 m. Figure 6 shows a histogram of the heights of the 58 buildings. Figure 5: Stabilization charts (circles denote poles determined by the stochastic subspace identification (SSI) method and solid lines denote the average power spectral density of the measured acceleration data): (**a**) Harbor View 1502 and (**b**) Central Park 103. Figure 7 compares the results obtained using the empirical formulae provided in KBC 2009 and ASCE 7-10 with the measured natural frequencies. Notably, the empirical formulae given in ASCE 7-10, which were derived through finite element analysis, yielded a natural period that was twice as long as the measured natural period. The natural period estimated using the formula given in KBC 2009 was relatively closer to the measured natural period, even though this formula pertains to the seismic design of RC moment frames. However, it can be seen from Figure 7 that the formula provided in KBC 2009 overestimates and underestimates, respectively, the natural periods of buildings shorter than 100 m and buildings taller than 200 m. This is because this formula is non-linearly proportional to building height with an exponent of 0.75, while the measured natural period is more or less linearly proportional to building height. Therefore, it can be concluded that the formulae provided in KBC 2009 and ACSCE 7-10 are unsuitable for use in the early stages of wind design. Figure 6: Histogram of heights of the 58 buildings.