Page 123 - Dynamic Loading and Design of Structures

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The foregoing linearized analysis is generally sufficient for civil engineering cases, but for
flexible systems tolerant of very large displacements, it may be necessary to estimate limiting
amplitudes taking account of the full pattern of CL (or Cz) as a function of incidence. Figure
3.6(b) shows a graphical construction of the force displacement loop. Starting from a
,
postulated amplitude ŷwith corresponding a, consideration of successive pairs of values of ÿ
and y leads to a plot which may comprise both energy input (continuous shading) and
dissipation (broken-line shading). The limiting amplitude is found by trial and error such that
the net input balances the dissipation by damping. Closed form algebraic procedures have also
been presented, commencing with a polynomial curve fit for C (Parkinson, 1965, Novak,
z
1972). Although the dominant parameters are normalized in the same grouping as for the
vortex shedding phenomenon, the resulting behaviour patterns are distinct:
●vortex shedding—critical speed VRC fixed, response amplitudes sensitive to Ks;
●galloping—critical speed V RC proportional to K , amplitudes likely to rise to much larger
s
values than typical of vortex shedding when V >V RC
R
Unfortunately interactions between these mechanisms of excitation commonly distort the
clarity of interpretation. Figure 3.7 shows three rectangular prisms tested as part of the
Department of Transport study (Wyatt and Scruton, 1981) undertaken to support the UK
Design Rules (BD49; Smith and Wyatt, 1981). The first case (deck width B equal to the depth
=
d) shows vortex shedding at VR 7 and galloping fairly distinct at perhaps VR=0.5K s. The third
case (B=3d) shows very clear vortex shedding at VR=10, but no evidence of galloping within
the range of the tests (V <1.5K ). The intermediate case clearly has some characteristics of
R
s
both mechanisms, strongly modified. To show these values in perspective, a steel box (e.g. a
bridge girder during erection) B=2d, plate thickness d/150 (plus allowance 50 per cent to
mass to allow for stiffeners, transverse elements, etc.) and damping log dec 0.03, would have
Ks approximately 20.

3.3.2 Flutter of bridge decks
An aerofoil, whether a flat plate or a slender smooth outline prism, does not show the
‘negative lift slope’ which is the key to galloping. However, violent self-excited oscillation of
aircraft wings has long been recognized as a potential hazard, under the name ‘flutter’.
Analysis based on the aerofoil flutter model has proved remarkably useful for slender bridges.
This proves to be essentially a coupling phenomenon, combining modes of vibration which in
still air are quite distinct, and dependent on the departure of flow patterns and resulting forces
from the quasi-steady model. This departure is not only a question of magnitude, but also of
phase shift between motion and force. A common method of description is by defining
coefficients for the force components proportional to instantaneous values of the rate of
change of the displacements as well as to the displacements themselves; these ‘derivative
coefficients’ are discussed further in Section 3.3.4.