Page 78 - Marine Structural Design

54 Pari I Structural Design Principles

Therefore, if possible, both short- and long-term approaches should be used to achieve a
conservative design.

3.5.2 Short-Term Extreme Approach
The short-term extreme values can be estimated based on a known initial probability
distribution of the maxima. For a Gaussian random response with zero-mean, the probability
density function of the maxima (peak values) can be represented by the following Rayleigh
distribution:

X X2
p(x) = -exp(--) x20 (3.4)
mo 2%
based on the assumption of a small bandwidth E, where,

mo, m2, and 1114 are the moments of response spectral density functions of zero-th, second, and
fourth order, respectively.
The cumulative probability distribution is:

The Probable Extreme Value (PEV) can be determined by:
XPEV = JGd&

Sometimes, the extreme response that is exceeded at a small possibility level a (risk
parameter), can be expressed as (Bhattacharyya, 1978):

where N, is the number of observations (or cycles) and xp~v represents the value that may be

exceeded once out of N observations. The chance for xp~y to be exceeded is lla times of
that for xe.& to be exceeded. a(l1) is chosen at the designer’s discretion, depending on the
condition of application.
For a response spectrum with a finite E, the probability density function of maxima in Eq. (3.4)
can be represented as (zhao, Bai & Shin, 2001):

r2
in which b(r) = eM--)dr .
%I%--
2
Similar to Eqs. (3.6) and (3.7, the PEV of responses is given by: