Page 97 - Dynamic Loading and Design of Structures

Page 74

The Fourier transform of the autocovariance (eqn (3.9)) gives the spectrum of the modal
generalized force as

(3.13)

Noting that the mean value is , this is commonly expressed in the form

(3.14)

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in which the correlation transfer function, or ‘aerodynamic admittance’, J , is

(3.15)

(Davenport, 1962; Bearman, 1981; Dyrbye and Hansen, 1997, Section 6.4.3). The integrals
comprise the whole structure. For a uniform slender horizontal structure, γz) reduces to the
(
mode shape function.
The foregoing development has presumed homogeneous wind (V and gust parameters σ u
and T s invariant with location on the structure). Variation in the input wind parameters,
generally the case with vertical structures, greatly increases algebraic complexity if
approached rigorously (Wyatt, 1981), but in practice it is usually sufficient to use constant
values, evaluated for a reference height selected by judgement (e.g. three-quarters of the
height of a tower or chimney). In this event, Pj for insertion in eqn (3.14) should be evaluated
consistently.
The full sequence of the spectral analysis is shown by Figure 3.2:
(a) the wind spectrum in the universal form (eqn (3.2)), defined on abscissa ñ=12nT , is
s
multiplied by
(b) the aerodynamic admittance expressing spatial correlation, reflecting H/Ln, defined here
on abscissa nH/ V, where H is the size of the structure (loaded length); the product
2
(a)×(b)×(2σ/V) gives the spectrum of the modal generalized force (normalized on the
u
mean value Pj);
(c) which is divided by the square of the modal generalized stifmess (Kj) and multiplied by
the square of the steady state dynamic magnifier (i.e.
in which δis the damping (as log dec) and nj is the natural
frequency (abscissa n/nj);
(d) to obtain the spectrum of modal generalized displacement SY.

The lowest natural frequency of any given structure is broadly predictable from its size.
Buildings typically have n =46/H, approximately. Towers and bridges are particularly well
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defined, given definition of the structural form and basic geometric proportions (chapter 3 in
Maguire and Wyatt, 1999). Consider a tower of height H=100m, which is likely to have a
frequency of about 0.67 Hz. A