UR
R0.1 0.995
AA
RA
SSYU
YU
YU
(6.28)
and finally the components of the vector X are determined as:
(0.1 0.995 )
AAAA
RA RR
SSSSXU
XU U
XU
(6.29)
whereby the limit state function can be written in terms of the uncorrelated and normalised
random variables U as follows:
gu( ) ((0.1 uARRRAAASS0.995 u) )(u ) (u S) (6.30)from which the reliability index can be calculated as in the previous example.
In case that the stochastically dependent basic random variables are not Normal or Log-
normal distributed the dependency can no longer be described completely in terms of
correlation coefficients and the above-described transformation is thus not appropriate. In
such cases other transformations must be applied as described in the next section.
6.7 Non-Normal and Dependent Random Variables
As stated in the previous the joint probability distribution function of a random vector X can
only be completely described in terms of the marginal probability distribution functions for
the individual components of the vector and the correlation coefficient matrix when all the
components of are either Normal or Log-normal distributed.
X
X
In the following consideration is first given to the simple case where the components of are
independent but non-Normal distributed. Thereafter it shall be seen how in some cases the
situation of jointly dependent and non-Normal distributed random variables may be treated.
X
The Normal-tail Approximation
One approach to consider the problem of non-Normal distributed random variables within the
context of the iterative scheme given in Equations (6.16)-(6.17) for the calculation of the
reliability index is to approximate the real probability distribution by a Normal probability
distribution in the design point.
As the design point is usually located in the tails of the distribution functions of the basic
random variables the scheme is often referred to as the “normal tail approximation”.
Denoting by x* the design point the approximation is introduced by:
- () (* i)
ii
i
iX
Xi
Xx
Fx