Four Problem Types

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Four Problem Types

In the probabilistic analysis, the problem is defined by a set of limit-state functions expressed in terms of a set of random variables denoted as x. The limit-state functions, based on the physical property, are further grouped into a mathematical model of cut sets. Each cut set represents a set of limit-states whose joint excess constitutes the failure of the system. According to the number of cut set and number of limit-state functions, UNIPASS™ can perform probabilistic analysis for four different problem types as described below.

In the most general case, the probability of failure of a system may be defined in the form

(1)

g_j(x) is the j-th limit-state function. C_k denotes the k-th cut set representing a set of limit-states whose joint excess (i.e. whose intersection ) constitutes the failure of the system, and the union is over all of the cut sets. For the special class of a series systems, C_k contains a single element for each cut set k, in Eq. (1). For the special class of parallel systems, there is only a single cut set, i.e., k=1 in Eq. ( 1).

UNIPASS^TM offers the following four problem types to cover any physical models.

Component Problem
Series System Problem
Parallel System Problem
General System Problem

Component Problem

A component problem has one cut set containing only one limit-state function, denoted as g(x).

Figure 1. Illustration of A Component Problem

Series System Problem

A series system problem is defined by two or more cut sets where each cut set contains a single limit-state function. The failure domain of an example defined by is illustrated in Figure 2.

Figure 2. Illustration of A Series System Problem

Parallel System Problem

A parallel system problem analyzes a system containing a single cut set that comprises more than one limit-state function. The failure domain for a parallel system is illustrated by an example defined as in Figure 3.

Figure 3. Illustration of A Parallel System Problem

General System Problem

A general system problem is a combination of serial systems and parallel systems. The failure domain for a general system problem is illustrated by an example defined as in Figure 4.

Figure 4. Illustration of A General System Problem

You should note that every row in the Cut Set list represents a cut set. Therefore, for a parallel system with series systems as its members, you should reformat the system with minimum cut sets, as in this example:

Figure 5. Example of Reforming A General System Problem in Terms of Minimum Cut Sets

Last Updated 11/12/08