Titel_SS06

appropriate curve has its minimum. From Figure 8 it is seen that if the cost of one test exceeds
1/200 of the initial manufacturing costs then there will be little to achieve by testing. For the
other cases it is seen that the minimum is generally quite flat and that the minimum tends to
be located in the range of 1-10 tests.

12.6 Typical Problems in Assessment and Maintenance

In the following a number of typical situations are considered with relation to assessment and
maintenance of existing structures. The treatment of the problems which I illustrated by
simple examples follows the approaches outlined in the foregoing.

Example 12.4 – Reliability updating by material strength testing

As an example considering updating of random variables consider the probabilistic modelling
of the yield stress of the steel bar. The prior probabilistic model for the yield stress of the steel
bar was assumed to normal distributed with known (deterministic) standard deviation
fyequal to 17.5 MPa and uncertain mean value. The mean value fywas assumed normal

distributed with known mean value @ 350 MPa and standard deviation @ 10 MPa.

Assume now, that 5 tests of the yield stress are performed on steel samples taken from a batch
of the same steel material. The test results are fˆy(365, 347, 354, 362, 348).

Based on the test results the prior probabilistic model for the mean value of the yield stress
can be updated using natural conjugate distributions as mentioned earlier.

In the case considered with a normally distributed variable with uncertain mean and known
standard deviation the posterior as given in Equation (12.1) may be found to reduce to
(Ditlevsen and Madsen (1996)):
2
()^11 exp
 2 2



-

 

y

fyy

f

f

! ! @@

" "

@@ "#" @@

$%$%

# (12.12)

where:

'

'

11

'





x

nn

nn

@@

(12.13)

and

(^22)
2

'

'



2







fy

nn

nn

@



@@

@@

(12.14)

and