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Reliability Block Diagrams

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Title: Reliability Block Diagrams


1
Reliability Block Diagrams
  • A reliability block diagram is a success-oriented
    network describing the function of the system.
  • If the system has more than one function, each
    function is considered individually, and separate
    reliability block diagram is established for each
    system function.
  • Each component is illustrated by a block. When
    there is a connection between the end points, we
    say that component i is functioning.

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Example
  • Consider a pipeline with two independent safety
    valves that are physically installed in series.
    These valves are supplied with a spring-loaded,
    fail-safe, close by hydraulic actuator. The
    valves are opened and held open by hydraulic
    control pressure and is closed automatically by
    spring force whenever the control pressure is
    removed or lost. In normal operation both valves
    are held open. The main function of the valves
    is to act as a safety barrier, i.e., to close and
    stop the flow in the pipeline in case of an
    emergency.

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Example
  • It is usually an easy task to convert a fault
    tree to a reliability block diagram. In this
    conversion, we start from the top event and
    replace the gate successfully. An OR-gate is
    replaced by a series structure of the
    components directly beneath the gate, and an
    AND-gate is replaced by a parallel structure of
    the components directly beneath the gate.

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Structure Function
  • The state of component i can be described by a
    binary state variable, i.e.,
  • Similarly the state of a system can be described
    by a binary function

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Series and Parallel Structures
  • Series
  • Parallel

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k-out-of-n Structure
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2-out-of-3 Structure
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Coherent Structures
  • Definition A system is said to be coherent if
    all its components are relevant and the structure
    function is non-decreasing in each argument.
  • Relevant
  • Irrelevant
  • Non-decreasing structure function

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Definitions
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Example
  • Component 2 is irrelevant

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2
a
b
1
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Some Theorems for Coherent Structures
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Redundancy at System Level
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Redundancy at Component Level
We obtain a better system by introducing
redundancy at component level than by introducing
redundancy at system level.
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Path Sets and Cut Sets
  • A structure of order n consists of n components
    numbered from 1 to n. The set of all components
    is denoted by C.
  • A path set P is a set of components in C which by
    functioning ensures that the system is
    functioning. A path set is said to be minimal if
    it cannot be reduced without loosing its status
    as a path set.
  • A cut set K is a set of components in C which by
    failing causes that the system to fail. A cut
    set is said to be minimal if it cannot be reduced
    without loosing its status as a cut set.

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Example 1
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Example 2
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Structures Represented by Paths
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Example 2
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Structures Represented by Cuts
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Example 2
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Critical Path
  • A critical path vector for component i is a state
    vector
  • Such that
  • A critical path set corresponding to the critical
    path vector for component i is defined by

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Structural Importance
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Example
  • Consider 2-out-of-3 structure

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Example

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Pivotal Decomposition
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Example Bridge Structure
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Structure of Composed Components
  • Partition into subsystems is done in such a way
    that each component never appears within more
    than one of the subsystems.

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Some Notations
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Coherent Modules
  • Let the coherent structure be given,
    and let
  • Then is said to be a coherent module
    of
  • if can be written as a function
  • where is the structure function of a
    coherent system.
  • What we actually do here is to consider all the
    components with the index belonging to A as one
    component with state variable . When
    we interpret the system in this way, the
    structure function can be written as

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Example
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Modular Decomposition
  • A modular decomposition of a coherent structure
    is a set of disjoint modules together with an
    organizing structure such that

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Prime Module
  • A module that cannot be partitioned into smaller
    modules without letting each component represent
    a module, is called a prime module.
  • III represents a prime module. II is not since it
    may be described in Fig 3.35 and hence can be
    partitioned into IIa and IIb.

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