## 4 Time Dependence of Wind Induced Floor Acceleration
### Time Dependence of the Floor Acceleration Correlation Coefficient  
The space correlation of two random processes is usually taken as a measure of how much one variable is related to another variable. In the case of a high-rise building, the correlation of floor acceleration at different building levels is a measure of the effects of aerodynamics, such as vortex shedding, and aeroelastics, such as nonlinear structural movements. Figure 8: Cumulative density function (CDF) of Mode #1 and \(\mathrm{z}/\mathrm{H}=1\) of the eigenvector with \(U=49.0\) along wind (**a**) and across wind (**b**) and \(111.8\) m/s along wind (**c**) and across wind (**d**). Figure 7: Standard deviation variation for Mode #1 along wind (**a**) and across wind (**b**). A further attempt was made to calculate a measure of response variability due to different acquisition records, examining the correlation coefficients \(\rho_{(6,1...5)}\) between the signal on the top, level #6 and on the other five levels, #1, #2, #3, #4 and #5, corresponding to z/H = 1/6, 2/6, 3/6, 4/6 and 5/6, for each 10 min recording session. Figure 9 shows the variability of the correlation coefficient \(\rho_{(6,1...5)}\) estimated for fifteen 10 min recording sessions with two significant velocities (\(U\) = 49.0 m/s and 111.8 m/s) and two wind angles (0\({}^{\circ}\) and 90\({}^{\circ}\)). The variability of \(\rho\) ranges from 5% to 75% and consequently is non-negligible. Its variability is larger along wind (Figure 9a) than across wind (Figure 9b) and increases with increases in velocity. It can be seen that the best along wind correlation for all velocities was calculated between levels #6 and #1, while the best across wind correlation was between levels #6 and #4. The correlation between two adjoining intermediate floors is quite low for all velocities and wind directions. This seems to suggest that along wind aerodynamics, in the range between z/H = 0.33 and z/H = 0.83, gives significant vorticity that induces a structural vibration different from that at the top. This effect is probably due to the not slender shape of the building, to the slabs and to the free vibration associated with the first structural modal shape which, in this case, is not a pendulum motion like that of a simple harmonic oscillator. The standard deviation of the \(\rho\) coefficient estimated for all fifteen along and across wind short (10 min) records ranges between 0.01 and 0.1 as shown in Figure 10. Figure 10 shows that the standard deviation, and consequently \(\rho\) variability, depends on flow velocity and on floor position. Values for the correlation between levels #6 and #5 along wind seem anomalous because the correlation is very small. It is reasonable to think that acquisitions were affected by noise during tests. The difference between small and large velocities is relevant but not the same for all levels. The three-dimensional trend shows a valley around  
Figure 9: Variability of the correlation coefficient from the building top (i.e., level #6) to levels #1, #2, #3, #4 and #5 along wind (**a**) and across wind (**b**). z/H = 0.5 for all velocities and two peaks at the top and close to the base for the highest velocities. This irregular trend of the standard deviation as a function of wind velocity and height above ground confirms that structural vibration is very different between the top and the other levels and also that the correlation coefficient is affected by the randomness of the wind flow distribution, consequently varying significantly from one time interval to another. The correlation coefficient between along and across wind floor acceleration was estimated, and its variability shown in Figure 11 for two significant velocities, \(U\) = 49.0 m/s and 111.8 m/s. It can be seen that the variability is slightly larger for \(U\) = 49 m/s than \(U\) = 111.8 m/s.