Titel_SS06

grmrBB(x) 1.563w

 2

(9.15)

By FORM analysis of the limit state Equations (9.14)-(9.15) the failure probabilities
and are readily calculated. As might already have been

anticipated on the basis of the elastic distribution of the moment on the beam the location A is
more critical than the location B.

-3

PF,AO9.58 10 ,B

-4

PF 4.56 × 10

The simple bounds for the failure probability of the series system may be found to be:

9.58 10^3 1 10

PF

 (9.16)

Defining failure not at level 1 but at level 2 is equivalent to defining failure as the formation
of a collapse mechanism. The formation of a plastic mechanism for the beam may in principle
occur in two different ways, either starting by the development of a plastic yield hinge at
location A and thereafter a plastic yield hinge in location B or the other way around, first at
location B and thereafter at location A. The block diagram to be considered in the systems
reliability analysis for the beam structure may thus be depicted as illustrated in Figure 9.9.

A

A"B

B

B"A

Figure 9.9: Block diagram for the systems reliability analysis of the beam structure at level 2 (or
mechanism level).

The probabilities for moment failure at location A and B have already been assessed
individually. Now first the case where failure is assumed to have taken place at location A
where a plastic yield hinge is formed is considered. The static system is thus changed as
illustrated in Figure 9.10, where the plastic moment capacity of the cross section at location A
has now been applied as a load counteracting rotation.

AB

F

10

R

W

R

Figure 9.10: Beam structure with yield hinge formed at location A.

For this static system a new limit state function for moment failure at location B may be
defined as: