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FIGURE 22.58 Strouhal number for a rectangular section.

22.5.3.2 Forces Due to Turbulent Flow

If the wind is turbulent, then the velocity of the wind along the wind direction is described as follows:

where J is the mean wind and u(t) is the turbulent component along the wind direction. The time dependent drag force per unit height is obtained from Equation 22.18 by replacing J by U(t). As the ratio u(t)/U is small, the time-dependent drag force can be expressed as fD(t )=J + fD(t) (22.24)

where JD and fD are the mean and the fluctuating parts of the drag force per unit height, which are given by

The spectral density of the fluctuating part of the drag force is obtained from the Fourier transformation of the autocorrelation function as

where Su(n) is the spectral density of the turbulent velocity and may be obtained from Equation 22.13.

In practice, the presence of the structure distorts the turbulent flow, particularly the small high-frequency eddies. A correction factor known as the aerodynamic admittance function w(n) may be introduced (Davenport 1961) to account for these effects. The following empirical formula has been suggested for w(n) (Vickery 1970):

where A is the frontal area of the structure. Now with the introduction of the aerodynamic admittance function, Equation 22.27 may be rewritten as

It is evident from Equation 22.26 that the fluctuating drag force varies linearly with the turbulence. Thus, large integral length scale and high turbulent intensities will cause strong buffeting and consequently increases the along wind response of the structure. However, the regularity of vortex shedding is affected by the presence of turbulence in the along wind and hence the across wind motion and torsional motion due to vortex shedding decreases as the level of turbulence increases.

### 22.5.4 Response Due to Along Wind

Tall slender buildings, where the breadth of the structure is small compared to the height, can be idealized as a line-like structure shown in Figure 22.59. Modeling the building as a continuous system, the governing equation of motion for along wind displacement x(z, t) can be written as (Heidebrecht and Smith 1973):

m(z)€(z, t) + c(z)X(z, t)+ EI(z)x""(z, t)- GA(z)x"(z, t) = f (z, t) (22.30)

where m, c, EI, GA are the mass, damping coefficient, flexural rigidity, and shear rigidity, per unit height, respectively. Furthermore, f(z, t) is the fluctuating wind load per unit height given in Equation 22.26.

FIGURE 22.59 Typical deflection mode of a shear wall-frame building.

FIGURE 22.59 Typical deflection mode of a shear wall-frame building.

Expressing the displacement in terms of the normal coordinates

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