Civil Engineering Reference
In-Depth Information
Wind-induced narrow-band vibrations can be taken to have a normal or Gaussian
probability distribution (Section C3.1). If this is the case then the peaks or amplitudes,
s,
have a Rayleigh distribution (e.g. Crandall and Mark, 1963):
(5.52)
where
σ
is the standard deviation of the entire stress history. Derivation of Equation
(5.52) is based on the level-crossing formula of Rice (1944-5).
Substituting into Equation (5.51),
(5.53)
Here the following mathematical result has been used (Crandall and Mark, 1963):
(5.54)
where Γ
(x)
is the Gamma Function.
Equation (5.53) is a very useful 'closed-form' result, but it is restricted by two
important assumptions:
• 'high-cycle' fatigue behaviour in which steel is in the elastic range, and for which an
s-
N
curve of the form of Equation (5.48) is valid, has been assumed;
• narrow-band vibration in a single resonant mode of the form shown in Figure 5.17 has
been assumed. In wind loading this is a good model of the behaviour for vortex-
shedding-induced vibrations in low turbulence conditions. For along-wind loading, the
background (sub-resonant) components are almost always important and result in a
random wide-band response of the structure.
5.6.3
Wide-band fatigue loading
Wide-band random vibration consists of contributions over a broad range of frequencies
with a large resonant peak—this type of response is typical for wind loading (Figure 5.7).
A number of cycle counting methods for wide-band stress variations have been proposed
(Bowling, 1972). One of the most realistic of these is the 'rainflow' method proposed by
Matsuishi and Endo (1968). In this method, which uses the analogy of rain flowing over
the undulations of a roof, cycles associated with complete hysteresis cycles of the metal
are identified. Use of this method rather than a simple level-crossing approach which is
the basis of the narrow-band approach described in Section 5.6.2, invariably results in
fewer cycle counts.
