The distribution of wind gusts in the atmosphere for flexible high-rise structures

Modern building standards are universal and applicable to the calculation of any structures with one or another measure of stock. However, the calculation of specific groups of structures requires detailed consideration and clarification. The article suggests some refinements in calculating of the high-rise structures made of metal for the pulsating component of the wind load.


Introduction
Wind load is a basic load for high-rise steel flexible structures such as towers, masts, steel chimneys. The behavior of such structures under load is described in a special way and strongly depends on the pulsations of the wind speed [1] - [8]. A more accurate determination of the mode of vibration of structures from a wind load as applied to flexible structures with a small vibration decrement can clarify and even reduce the value of the applied load.

Wind spectrum
First, consider the wind spectrum. Wind flow velocities, aerodynamic forces, and the reaction of structures to them are random functions of time and are expressed by the energy spectrum S (ω), which shows how much power of a given process falls on each infinitesimal frequency range dω. For longitudinal pulsations of the wind speed, the empirical Davenport spectrum [4] (underlying [6]) is used: where 10 /X fL u -reduced frequency; foscillation frequency, Hz; Llength scale; 10 Xaverage speed at altitude 10m; k оdrag coefficient of the underlying surface.
The spectral characteristics of aerodynamic forces and reactions can be obtained from the speed spectrum using the transfer functions χ (Fig. 1) [9]. The spectrum of the reaction of the structure to the effect of the random pulsation process is shown in Fig. 2 [4,9]. It clearly shows the components of the pulsations of the wind velocity: quasistatic (with a maximum of the energy spectrum of the velocity pulsations) and resonant (the maximum is the frequency of natural vibrations). The structures under consideration in the described ranges behave differently. Separation of the reaction into quasistatic and resonant allows avoiding errors in the calculations of fatigue strength and durability; to compensate for the error in pointing the antennas installed on the structures, individually for each component; install dynamic vibration dampers that work only in the resonant frequency region [2], [8], [10]. When determining the pulsation component in the formulas from SP 20.13330.2018 Loads and actions, the coefficient of pulsations of wind pressure at the level of z is used: where D сthe security ratio, in SP 20.13330.2018 is taken equal to 2.5; J tturbulence intensity is determined by the expression  The authors of the article propose separation not at the stage of reactions, but at the stage of application of the load [8], presenting the pulsation component as The ripple coefficient will be determined separately for quasistatic (p.q.st), separately for resonance (p.res.) by integrating over two ranges -from 0 to f' and from f' to ∞ (the value of the frequency f ' will mean a certain boundary of the ranges).

Frequency response
Under the influence of a gust of wind, the amplitude of the oscillation of the structure increases gradually (there is a transition process and the unsteady oscillations -A non-st ) and soon reaches a maximum -the steady-state amplitude А st (Fig. 3).

Fig. 3. The increase in the amplitude of oscillations
The duration of the transition process depends on the decrement of oscillations [5]: the number of oscillation cycles until А st is reached, the greater, the smaller δ. To achieve А st , tower structures with δ = 0,10 must be subjected to 13 cycles of successive gusts of wind of the same intensity equal to 2,5σ υ' formula (2). However, in reality, only the rush of the first cycle has such intensity, after which the intensity decreases. Thus, the calculated amplitude can be taken as the amplitude corresponding to 1 cycle of oscillation (2 or 3 cycles -to safety margin) -i.e. at the beginning of the transition mode.
The equation of motion of the system will look like: where n -is the attenuation index; h -is the amplitude of the reduced external force; Z -frequency of external force; Z o is the frequency of natural vibrations of the structure.
Frequency response (AFC) for a steady process of oscillations (from (6)): where βcoefficient equal to Frequency response of an unsteady process: The dynamic coefficient (SP 20.13330.2018) used to determine the pulsation component for the steady state ξ is proportional to (7), for the unsteady one it is proportional to (9). Then

Distribution in the atmosphere
Currently, when developing models of the dynamic effects of wind gusts on structures, it is assumed that wind gusts in the atmosphere are distributed according to the normal law. The third and fourth static moments of the distribution, reflecting the asymmetry and excess, are equal to zero. The influence of positive (coinciding with the direction of the average wind flow) and negative (directed towards the average wind flow) gusts is equally probable, which allows us to consider the effect as sinusoidal, described by the Lagrange equation (6) for a system with one degree of freedom. Numerous meteorological studies conducted in recent years on high-rise towers, masts, and wind tunnels [7] have shown that the laws of the distribution of wind gusts are significantly different from normal. So, for example, at the lower levels (0-100m) of the atmospheric boundary layer, there is a positive, and at the upper levels (100-300m) negative asymmetry of the laws of distribution of wind gusts. This makes it possible to replace the model described by equation (6) (11) for the lower levels of the atmospheric boundary layer, and for the upper levels:

Conclusions
The calculation of structures for wind gusts according to models (6) and (11)(12) shows that in the second case, the calculated amplitudes of movement of the structures are much smaller than in the first case, assuming the law of the distribution of wind gusts in the atmosphere is normal. It is advisable to use these results when developing new and improving existing regulatory documents.