Page 238 - Dynamic Loading and Design of Structures

Page 209

Weibull distribution with γ2 and is the variance of the response). In a long-term
=
,
p
period τthe number of (narrow-band) cycles associated with sea state i, is ni=τ ivi, where pi
is the long-term probability and the number of cycles per time unit, respectively, of this sea
i
state. Hence the cumulative damage in τs

(5.58)

where ρis a correction factor to account for wide-band and/or non-Gaussian load effects.
i
Stress ranges due to wide-band Gaussian or non-Gaussian response processes should be
determined by an appropriate method of cycle counting (e.g. the rainflow method, see chapter
4 of Almar-Næss, 1985). Simple, conservative methods for combining high and low
frequency responses may be applied. Fatigue damage may be calculated by assuming that the
number of cycles is determined by the zero-upcrossing frequency and that the distribution of
stress ranges follow a Rayleigh distribution. Wirsching and Light (1980) established an
empirical correction to the fatigue damage determined by the narrow-band assumption.
Extensive evaluations of various empirical, closed form methods for correct-ing the fatigue
damage obtained by the narrow-band approach, show that Dirlik’s (1985) method yields the
best estimates. Jiao and Moan (1990) analytically derived a correction factor which yields
reasonable estimates.
Leira et al. (1990) demonstrate that accurate fatigue estimates can be obtained for cases
with non-linear effects by establishing a quasi-transfer function H(ω) that is used to calculate
the response for all sea states, and is defined by is obtained by
calculating S (ω) using time domain samples of response for a sea state with spectral density
x
Sζω). The significant wave height of this sea state is given by
(

(5.59)

where m is the exponent in the SN-curve, pi is the relative frequency of a sea state number i

and w is a weight function Dav being the average diameter of
i
loaded structural members. Even if the statistical uncertainty is less for the response relevant
to fatigue than for extreme response, a sufficient sample to limit this uncertainty should be
used. The load effects are described by a Weibull fit to the stress range distribution. The
location and scale parameters are expressed by the standard deviation (Farnes and Moan,
1994). Hence, the relevant location and scale parameters for other sea states can be obtained
when the variance is determined from the frequency domain results.
Equation (5.57) applied for a period τwith nτ =n0 cycles is convenient as a basis for
discussing the sensitivity of fatigue damage to various parameters. The shape factor γof the
Weibull distribution then depends on environmental conditions,