Page 443 - Marine Structural Design

Chapter 23 Basics of Structural Reliabiliry 419

23.3.4 Calculation by FORM

In the First Order Reliability Method (FORM) the failure set is approximated by first
transforming the limit state surface into U space and then replacing it by its tangent hyperplane
at the design point u*.
The Rosenblatt transformation is used,
(23.15)

If the random variables Z are mutually independent, the transformation is simply
ui = w(F~(z~)) i = 1,2 ,....., n (23.16)
The limit state surface g(Z)=O in Z space is transformed into a corresponding limit state
surface g(u)=O in U space. In a next step, the design point has to be determined in U space.
This points lies on g(u)=O, and is the point in the failure set with the largest probability density,
that is, the closest point on the failure surface to the origin of the U space. An interactive
procedure is applied to find the design point u*, which is expressed as
u' =pa* (23.17)

in which p is the first order reliability index, or the distance between the design point and the
origin. The unit normal vector a* to the failure surface at u* are calculated by

(23.18)

where Vg(u) is the gradient vector. The actual limit state surface &)TO is then approximated
by its tangent hyperplane, which has the equation at the design point u as
g(u)=p+aTu=o (23.19)

The first order safety margin M is defined as
M = g(U) = P+aTU (23.20)
The corresponding approximation to the failure probability is
Pf = a(- P) (23.21)

From the above descriptions of FORM, it is seen that FORM is to approximate the failure set
through replacement of the limit state surface in u-space by its tangent hyperplane at the
design point, as defined by Eq. (23.19). Figure 23.3 illustrates this method.
This is the first order approximation to the failure probability Pf, and p is the corresponding
first order approximation to the reliability index. The accuracy of the estimate depends upon
how well the true failure surface is represented by the linear approximation, and may usually
be improved by SORM, which described in the following sub-section. The problem which
affects the accuracy most is that FORM may not be able to find the global design point in
cases of multiple design points.