Page 473 - Marine Structural Design

Chapter 25 Reliability of Ship Structures 449

g(xlt) = CUMU 0) - c, VSM, (4 + M,(Ol (25.23)
where M,,(f) is the ultimate strength, M&) and Mdt) are the stillwater and wave-induced
moments respectively; C,, and C, represent the model errors in predicting the hull's ultimate
strength and combined total bending moment the ship experiences. The failure probability at
the time Tis expressed by
'I g(J+)<O 1

pf(T) = 5 5.- If. (xlt)dx fT(t)dt (25.24)
0
where fx (xlt) is the joint probability density function; fT (t) is the probability density function
of occurrence time T, which is assumed as a uniform distribution, fT (t) = I/T . Therefore,
Eq.(25.24) can be rewritten by

(25.25)

By defining P,(t) as a conditional failure probability at time 2,

P, 0) = J.. Jfx (xlw (25.26)
g(xlf)<O
the failure probability has in a simple form

(25.27)

The response surface method (Bucher, 1990) is applied when the limit state function g(r1t)is
expressed implicitly and has a nonlinear form in order to overcome the expensive
computational effort integrating Eq. (25.25) in evaluating the failure probability.
The basic concept of the response surface method is to approximate the original complex
and/or implicit limit state function using a simple and explicit function. The accuracy of the
results depends highly on how accurately the characteristics of the original limit state are
represented by approximate function. The suitability of the response surface obtained relies
mainly on the proper location of so-called sampling points. Many algorithms have been
proposed to select appropriate sampling points, which promise to yield better response surface
fitting. In addition, the basic function shape is also known to be another major factor that
influences both the accuracy of the response surface method and the selection of the reliability
evaluation method.
Many practical reliability evaluation techniques are available once the failure surface G(x) is
defined in an explicit closed form. Among those techniques, first order method is commonly
used due to the efficiency with acceptable accuracy. An equivalent linearized limit state at so-
called design point is taken and the safety margin of the structural system is determined as the
minimum distance from the origin to the original nonlinear limit surface in the independent
standard normal space.