Titel_SS06

AB

W

10

Figure 9.7: Simple beam structure subjected to a point load F.

The -unzipping Method

According to the -unzipping method failure may be defined at level where refers to the
number of failures in the system, which is associated with failure of the system. E.g. one can
define failure of a structural system as the event that one failure mode has occurred but it is
also possible to define failure of the system as the event that two or more ( ) failure modes
have occurred. The maximum number of failure modes, which can be considered depends on
the number of failure modes required for the formation of a collapse mechanism of the
structural system.

nn

n

As already stated moment failure of the beam is the concern, and as already known, according
to the theory of elasticity, the moment distribution on the beam has a minimum at location A
equal to -1.875Wand a maximum at location B equal to 1.563W moment failures at
location A and B are considered as the potential failure modes of the system used to describe
the reliability of the beam.

Defining the failure for the structural system as the event of failure of any one of the
considered failure modes for the system – a level 1 reliability analysis - the systems reliability
analysis may be performed by consideration of the series system illustrated in Figure 9.8.

A B

Figure 9.8: Block diagram used for the structural reliability analysis of the beam structure at level 1.

For the general case where the considered structural system may include many potential
failure modes the number of failure modes to be taken into account may be limited by

considering only those failure modes with reliability indexes in the interval min,min (^2) i
where min is the smallest reliability index for the considered failure modes and 2 i is an
appropriately selected constant which defines the total number of failure modes to be taken
into account when analysing the considered system at level i.
The limit state functions for the moment failure modes at the two locations A and B may be
written as:
grmrAA(x) 1.875w (9.14)