Titel_SS06

M

M







 (6.9)

The reliability index  as defined in Equation (6.9) has a geometrical interpretation as
illustrated in Figure 6.1 where a two dimensional case is considered:

-2

0

2 4 6 8 10 12

-2

-4

-6

2

4

6

8

10

12

0

S

R

x 2

x 1

g (x) = 0

-2

0

(^246810)
-4
-6
2
4
6
8
10
12
0
S
R
u 2
u 1
g (u) = 0
ß
12
Figure 6.1: Illustration of the two-dimensional case of a linear limit state function and standardised
Normal distributed variables U.
In Figure 6.1 the limit state function has been transformed into the limit state function
by normalisation of the random variables in to standardized Normal distributed random
variables as:
g()x
g()u
i
i
iX
i
X

X

U







 (6.10)

such that the random variables Ui have zero means and unit standard deviations.

Then the reliability index  has the simple geometrical interpretation as the smallest distance

from the line (or generally the hyper-plane) forming the boundary between the safe domain
and the failure domain, i.e. the domain defined by the failure event. It should be noted that this
definition of the reliability index (due to Hasofer and Lind (1974)) does not depend on the
limit state function but rather on the boundary between the safe domain and the failure
domain. The point on the failure surface with the smallest distance to the origin is commonly
denoted the design point or most likely the failure point.

It is seen that the evaluation of the probability of failure in this simple case reduces to some
simple evaluations in terms of mean values and standard deviations of the basic random
variables, i.e. the first and second order information.

6.4 The Error Accumulation Law

The results given in Equation (6.6) have been applied to study the statistical characteristics of
errors ' accumulating in accordance with some differentiable function ( )f x , i.e.: