Page 110 - Dynamic Loading and Design of Structures

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supports. The dimensions must be expressed in metres. For chord members, cf is commonly
not much greater than unity, whereas for bracing members cf is typically about 1.9
(corresponding to EI/L values for the bracing around one-quarter of the value for the chord).
For full end fixity, cf=2.27. To achieve , the above equation gives the limiting
slenderness as

(3.35)

For a tubular bracing with m and cf=1.9, the favourable size number range
corresponds to . Significantly higher values of L/D than given by this condition will
lead to unfavourable subcritical resonance, which should be avoided unless a high value of K s
is assured, exceeding 20 or 25.
It has long been appreciated that motion of the structure led to locking of the shedding
frequency to the structural frequency over a range of reduced velocity extending from
marginally below the value given by the reciprocal of the stationary body Strouhal number to
a value typically some 30 per cent larger. Very small movements are sufficient, possibly as
low as ŷ=0.015D. As the coefficient of alternating lift can be sustained over much of this
range, the maximum response may be increased, occurring at a higher speed than given by the
stationary body VRC.
It is apparent, however, that this effect has an equally important action in encouraging
coherent excitation in the face of contrary factors, which include the variation of mean speed
over the height of a vertical or inclined structure, taper of the structure (giving a pro-rata
variation of the nominal resonance speed), gusts (continually varying local speeds) and indeed
the inherent randomness of the above-critical separated flow. A parametric study of chimneys
6
at Reynolds numbers up to 2×10 was carried out in the National Physical Laboratory
compressed air-wind tunnel (Wootton, 1969). It was noted that whereas reduction of the
Scruton number from 16 to 8 typically caused an increase of r.m.s. response tip displacement
from 0.01D to 0.015D (in line with prediction by the stochastic model described below
presuming relatively poor synchronization of shedding over the length of the structure),
further reduction to K =4 caused the displacement to rise sixfold, to more than 0.1D.
s
The lock-on effect is of special importance in promoting a well organized net excitation in
turbulent or sheared flow, although a larger amplitude may be required to achieve an equal
result. Noting that for the fundamental mode of a chimney, the majority of the modal
excitation is derived from (say) the top third of the height, lock-on is commonly presumed to
be effective over this length if its taper does not exceed about 20 per cent (±10 per cent on
mean diameter). Shear, as represented by the change in the mean speed, is generally less than
this (e.g. a power law with index 0.15 gives a variation of only ±3 per cent over the top third
of the height.

3.2.5 Stochastic modelling of vortex shedding

Ignoring, for the moment, the feedback of structural motion, the degree of randomness
inherent in boundary layer and wake effects suggests application to this