Page 92 - Dynamic Loading and Design of Structures

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simple concept that the spectrum defines the strength of an infinite number of infin-itesimally
spaced sinusoid components which are sustained so that the steady state response is attained
for each component. The essential randomness results from the infinite complexity of the beat
phenomena between such components. The magnitudes are defined in mean-square terms; the
spectrum describes the distribution of the variance (mean square deviation from the mean) of
2
the process on a frequency abscissa, and the ordinates are thus values of (process) per unit of
2
frequency. Throughout this chapter the notation σ( · ) is used for the variance of the quantity
(
indicated in the parenthesis (in the case of windspeed, σ u is used for σu) to facilitate concise
presentation).
Wind engineering is exceedingly fortunate that the choices of spectral definition and
notation in the seminal presentations (Davenport, 1961, 1962) have been universally followed.
The basic spectrum is used in the single-sided form with frequency (n) expressed in Hz,
which is denoted S(n) (although W(n) has become more common as the notation for this form
in other fields). The numerical values of ordinates in the wind engineering format are thus 4π
)
times the values for the double-sided circular-frequency form given as S(ω in eqn 10.20. It is
further general in wind engineering to present spectra in the normalized non-dimensional
2
format of nS(n)/σ plotted on a logarithmic scale abscissa n. Noting that

(3.1)

this preserves the visual interpretation of the area of the spectral plot as the variance of the
process. It gives a clear graphical representation despite the considerable frequency range
present in the natural wind and has the great convenience that scaling parameters applicable to
the frequency abscissa have the effect only of a ‘rigid body’ shift of the normalized shape.
The Harris—von Karman normalized form (Figure 3.2a, Section 3.1.3) for the alongwind
gust component (u) is

(3.2)

The frequency normalization favoured by the present author is ñ=12nT, in which T is the
timescale, the (one-sided) integral of the autocorrelation function of the windspeed. Theory
derived for Homogeneous Isotropic Turbulence (HIT), which ignores the distortion of the
turbulence field resulting from proximity to the ground (where w must clearly be zero), gives
the numerical factor as . The length scale
may al-ternatively be used as the input parameter for frequency normalization.
Dynamic analysis is focused on the upper tail of the spectrum ñ>>1, as demonstrated later; in
this range

(3.3)