Titel_SS06

batch as the considered steel bar and that the test result is fˆy=350 MPa i.e. equal to the mean

value of the prior probabilistic model offy. Then the likelihoods are the probabilities of the

observation ˆfy given the event of failure and survival, respectively.

The likelihoods corresponding to the situation where the steel bar is not strengthened i.e.
decision a 0 , are calculated using e.g. FORM/SORM analysis, see Madsen et al. (1986).







(^122)
0
2

ˆ 0

ˆ 1.66 10

0

:

yy y

y

y

Pf f f A s

Pf

Pf As

  













(^122)
1
2

ˆ 0

ˆ 1.98 10

0

:

yyy

y

y

ffAs

Pf

Pf A s

Pf 









where fy 1 and fy 2 are two different identical distributed random variables with distribution

function taken as the prior distribution for fy and with common parameters @ and @. A =

104 mm^2.

In the expressions for the calculation of the likelihoods the first event in the numerator is the
observation event. It is in this event where the modelling of the accuracy of the inspection or
test method must be included. In the above example no account of measurement uncertainty
was considered. Adding a random variable to the measured yield stress fˆycould have done

this. The more measurement uncertainty the weaker is the likelihood.

The posterior probabilities for the two states : 0 and : 1 may now be calculated using the prior

probabilities:

Pa@

: 00  11 .15 10^2 0.9885

and

Pa@

: 10 1.15 10 ^2

as



22

(^00222)

ˆ, 1.66 10 (1 10 ) 0.9905

(1.66 10 (1 1.15 10 1.98 10 1.15 )

Pfa: y



 

@@  

 ^2  10

1.15

)





(^222)
10 222  10
a 1
2

ˆ, 1.98 10^10 0.95 10

(1.66 10 (1 1.15 10 1.98 10 1.15 )

Pfa: y

 

@@  

1.15

)



By comparison with the prior probabilities it is readily seen that the test result has reduced the
probability of failure.

Now the posterior probabilities for the situation where the steel bar is strengthened by an
increase of the cross sectional area of 10% i.e. decision are considered. The calculations