Fault Tree Diagrams and System Analysis - Revision history

Sam Eisenberg: /* Trigger Event */

2019-11-27T21:25:00Z

Trigger Event

← Older revision		Revision as of 21:25, 27 November 2019
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	An undeveloped event has the same properties as a basic event with the exception that it is graphically rendered as a diamond. The diamond representation graphically illustrates that this event could have been expanded into a separate fault tree but was not. In other words, the analyst uses a different symbol to convey that the event could have been developed (broken down) further but he/she has chosen not to do so for the analysis.		An undeveloped event has the same properties as a basic event with the exception that it is graphically rendered as a diamond. The diamond representation graphically illustrates that this event could have been expanded into a separate fault tree but was not. In other words, the analyst uses a different symbol to convey that the event could have been developed (broken down) further but he/she has chosen not to do so for the analysis.

	===~~House~~ Event===		===Trigger Event===
	[[Image:pentagon.png\|center\|100px\|link=]]		[[Image:pentagon.png\|center\|100px\|link=]]

	A ~~house~~ event is an event that can be set to occur or not occur (i.e., it usually has a fixed probability of 0 or 1). It is usually used to turn paths on or off or to make paths of a tree functional or non-functional. Furthermore, the terms failed house and working house have been used to signify probabilities of 0 and 1 respectively. In BlockSim, a house shape is available for an event and a house event has the same properties as a basic event, keeping in mind that an event can be set to Cannot Fail or Failed from the block properties window.		A trigger event is an event that can be set to occur or not occur (i.e., it usually has a fixed probability of 0 or 1). It is usually used to turn paths on or off or to make paths of a tree functional or non-functional. Furthermore, the terms failed house and working house have been used to signify probabilities of 0 and 1 respectively. In BlockSim, a house shape is available for an event and a house-shaped event has the same properties as a basic event, keeping in mind that an event can be set to Cannot Fail or Failed from the block properties window.

	===Conditional Event===		===Conditional Event===

Richard House: /* Minimal Cut Sets */

2015-12-22T21:14:11Z

Minimal Cut Sets

← Older revision		Revision as of 21:14, 22 December 2015
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	Traditional solution of fault trees involves the determination of so-called ''minimal cut sets''. Minimal cut sets are all the unique combinations of component failures that can cause system failure. Specifically, a cut set is said to be a minimal cut set if, when any basic event is removed from the set, the remaining events collectively are no longer a cut set, as discussed in Kececioglu [[Appendix_B:_References \| [10]]]. As an example, consider the fault tree shown in the figure below. The system will fail if {1, 2, 3 and 4 fail} or {1, 2 and 3 fail} or {1, 2 and 4 fail}.		Traditional solution of fault trees involves the determination of so-called ''minimal cut sets''. Minimal cut sets are all the unique combinations of component failures that can cause system failure. Specifically, a cut set is said to be a minimal cut set if, when any basic event is removed from the set, the remaining events collectively are no longer a cut set, as discussed in Kececioglu [[Appendix_B:_References \| [10]]]. As an example, consider the fault tree shown in the figure below. The system will fail if {1, 2, 3 and 4 fail} or {1, 2 and 3 fail} or {1, 2 and 4 fail}.

	[[Image:9.png\|center\|250px\|Minimal cut set example.]]		[[Image:9.png\|center\|250px\|Minimal cut set example.\|link=]]

	All of these are cut sets. However, the one including all components is not a minimal cut set because, if 3 and 4 are removed, the remaining events are also a cut set. Therefore, the minimal cut sets for this configuration are {1, 2 , 3} or {1, 2, 4}. This may be more evident by examining the RBD equivalent of the figure above, as shown in the figure below.		All of these are cut sets. However, the one including all components is not a minimal cut set because, if 3 and 4 are removed, the remaining events are also a cut set. Therefore, the minimal cut sets for this configuration are {1, 2 , 3} or {1, 2, 4}. This may be more evident by examining the RBD equivalent of the figure above, as shown in the figure below.

	[[Image:BS10.png\|center\|350px\|RBD of the fault tree shown in figure above.]]		[[Image:BS10.png\|center\|350px\|RBD of the fault tree shown in figure above.\|link=]]

	BlockSim does not use the cut sets methodology when analyzing fault trees. However, interested users can obtain these cut sets for both fault trees and block diagrams with the command available in the Analysis Ribbon.		BlockSim does not use the cut sets methodology when analyzing fault trees. However, interested users can obtain these cut sets for both fault trees and block diagrams with the command available in the Analysis Ribbon.

Richard House: /* Fault Trees and Simulation */

2015-12-22T21:12:20Z

Fault Trees and Simulation

← Older revision		Revision as of 21:12, 22 December 2015
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	A sample table of simulation results is given next for up to <math>t=1,000\,\!</math>, using <math>2,000\,\!</math> simulations for each diagram and an identical seed.		A sample table of simulation results is given next for up to <math>t=1,000\,\!</math>, using <math>2,000\,\!</math> simulations for each diagram and an identical seed.

	[[Image:FT and RBD example1.png\|center\|400px\|]]		[[Image:FT and RBD example1.png\|center\|400px\|link=\|]]

	As expected, the results are equivalent (within an expected difference due to simulation) regardless of the diagram type used. It should be pointed out that even though the same seed was used by both diagrams, the results are not always expected to be identical because the order in which the blocks are read from a fault tree diagram during the simulation may differ from the order in which they are read in the RBD; thus using a different random number stream for each block (e.g., block ''G'' in the RBD may receive a different sequence of random numbers than event block ''G'' in the FT).		As expected, the results are equivalent (within an expected difference due to simulation) regardless of the diagram type used. It should be pointed out that even though the same seed was used by both diagrams, the results are not always expected to be identical because the order in which the blocks are read from a fault tree diagram during the simulation may differ from the order in which they are read in the RBD; thus using a different random number stream for each block (e.g., block ''G'' in the RBD may receive a different sequence of random numbers than event block ''G'' in the FT).

Richard House: /* Fault Trees and Simulation */

2015-12-22T21:11:52Z

Fault Trees and Simulation

← Older revision		Revision as of 21:11, 22 December 2015
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	As an example, consider the RBD shown in the first figure below and its equivalent fault tree representation, as shown in the second figure.		As an example, consider the RBD shown in the first figure below and its equivalent fault tree representation, as shown in the second figure.

	[[Image:10.44.png\|center\|450px\|RBD for a repairable system.]]		[[Image:10.44.png\|center\|450px\|RBD for a repairable system.\|link=]]


	[[Image:10.45.png\|center\|450px\|Fault tree equivalent of the repairable system shown in figure above.]]		[[Image:10.45.png\|center\|450px\|Fault tree equivalent of the repairable system shown in figure above.\|link=]]

	Furthermore, assume the following basic failure and repair properties for each block and event:		Furthermore, assume the following basic failure and repair properties for each block and event:

Richard House: /* Using Mirrored Blocks to Represent Complex RBDs as FTDs */

2015-12-22T21:11:16Z

Using Mirrored Blocks to Represent Complex RBDs as FTDs

← Older revision		Revision as of 21:11, 22 December 2015
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	A fault tree cannot normally represent a complex RBD. As an example, consider the RBD shown in the figure below.		A fault tree cannot normally represent a complex RBD. As an example, consider the RBD shown in the figure below.

	[[Image:10.41.png\|center\|400px\|A complex RBD that cannot be represented by a fault tree unless duplicate events are utilized.]]		[[Image:10.41.png\|center\|400px\|A complex RBD that cannot be represented by a fault tree unless duplicate events are utilized.\|link=]]

	A fault tree representation for this RBD is:		A fault tree representation for this RBD is:

	[[Image:10.42.png\|center\|350px\|A fault tree representation using mirrored blocks (events) of the complex RBD.]]		[[Image:10.42.png\|center\|350px\|A fault tree representation using mirrored blocks (events) of the complex RBD.\|link=]]

	Note that the same event is used more than once in the fault tree diagram. To correctly analyze this, the duplicate events need to be set up as "mirrored" events to the parent event. In other words, the same event is represented in two locations in the fault tree diagram. It should be pointed out that the RBD in the following figure is also equivalent to the RBD shown earlier and the fault tree of the figure shown above.		Note that the same event is used more than once in the fault tree diagram. To correctly analyze this, the duplicate events need to be set up as "mirrored" events to the parent event. In other words, the same event is represented in two locations in the fault tree diagram. It should be pointed out that the RBD in the following figure is also equivalent to the RBD shown earlier and the fault tree of the figure shown above.

	[[Image:10.43.png\|center\|550px\|An RBD using mirrored blocks that is equivalent to both the RBD and FTD.]]		[[Image:10.43.png\|center\|550px\|An RBD using mirrored blocks that is equivalent to both the RBD and FTD.\|link=]]

	=Fault Trees and Simulation=		=Fault Trees and Simulation=

Richard House: /* Conditional Event */

2015-12-22T21:02:02Z

Conditional Event

← Older revision		Revision as of 21:02, 22 December 2015
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	A conditional event is represented by an ellipse and specifies a condition. Again, it has all the properties of a basic event. It can be applied to any gate. As an example, event <math>C\,\!</math> in the first figure below would be the conditional event and it would be represented more applicably by an ellipse than a circle, as shown in the second figure below.		A conditional event is represented by an ellipse and specifies a condition. Again, it has all the properties of a basic event. It can be applied to any gate. As an example, event <math>C\,\!</math> in the first figure below would be the conditional event and it would be represented more applicably by an ellipse than a circle, as shown in the second figure below.

	[[Image:10.10.png\|center\|200px\|Using an AND gate to represent an inhibit relationship.link=]]		[[Image:10.10.png\|center\|200px\|Using an AND gate to represent an inhibit relationship.\|link=]]

	[[Image:10.22.png\|center\|200px\|Using an ellipse attached to an inhubit gate (with no gate properties) to show the conditional event. This is mathematically equivalent to figure above.]]		[[Image:10.22.png\|center\|200px\|Using an ellipse attached to an inhubit gate (with no gate properties) to show the conditional event. This is mathematically equivalent to figure above.\|link=]]

	=Comparing Fault Trees and RBDs=		=Comparing Fault Trees and RBDs=

Richard House: /* Conditional Event */

2015-12-22T21:01:32Z

Conditional Event

← Older revision		Revision as of 21:01, 22 December 2015
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	===Conditional Event===		===Conditional Event===
	[[Image:oval.png\|center\|100px\|]]		[[Image:oval.png\|center\|100px\|link=]]

	A conditional event is represented by an ellipse and specifies a condition. Again, it has all the properties of a basic event. It can be applied to any gate. As an example, event <math>C\,\!</math> in the first figure below would be the conditional event and it would be represented more applicably by an ellipse than a circle, as shown in the second figure below.		A conditional event is represented by an ellipse and specifies a condition. Again, it has all the properties of a basic event. It can be applied to any gate. As an example, event <math>C\,\!</math> in the first figure below would be the conditional event and it would be represented more applicably by an ellipse than a circle, as shown in the second figure below.

	[[Image:10.10.png\|center\|200px\|Using an AND gate to represent an inhibit relationship.]]		[[Image:10.10.png\|center\|200px\|Using an AND gate to represent an inhibit relationship.link=]]

	[[Image:10.22.png\|center\|200px\|Using an ellipse attached to an inhubit gate (with no gate properties) to show the conditional event. This is mathematically equivalent to figure above.]]		[[Image:10.22.png\|center\|200px\|Using an ellipse attached to an inhubit gate (with no gate properties) to show the conditional event. This is mathematically equivalent to figure above.]]

Richard House: /* House Event */

2015-12-22T21:01:13Z

House Event

← Older revision		Revision as of 21:01, 22 December 2015
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	===House Event===		===House Event===
	[[Image:pentagon.png\|center\|100px\|]]		[[Image:pentagon.png\|center\|100px\|link=]]

	A house event is an event that can be set to occur or not occur (i.e., it usually has a fixed probability of 0 or 1). It is usually used to turn paths on or off or to make paths of a tree functional or non-functional. Furthermore, the terms failed house and working house have been used to signify probabilities of 0 and 1 respectively. In BlockSim, a house shape is available for an event and a house event has the same properties as a basic event, keeping in mind that an event can be set to Cannot Fail or Failed from the block properties window.		A house event is an event that can be set to occur or not occur (i.e., it usually has a fixed probability of 0 or 1). It is usually used to turn paths on or off or to make paths of a tree functional or non-functional. Furthermore, the terms failed house and working house have been used to signify probabilities of 0 and 1 respectively. In BlockSim, a house shape is available for an event and a house event has the same properties as a basic event, keeping in mind that an event can be set to Cannot Fail or Failed from the block properties window.

Richard House: /* Undeveloped Event */

2015-12-22T21:01:00Z

Undeveloped Event

← Older revision		Revision as of 21:01, 22 December 2015
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	===Undeveloped Event===		===Undeveloped Event===

	[[Image:diamond.png\|center\|100px\|]]		[[Image:diamond.png\|center\|100px\|link=]]

	An undeveloped event has the same properties as a basic event with the exception that it is graphically rendered as a diamond. The diamond representation graphically illustrates that this event could have been expanded into a separate fault tree but was not. In other words, the analyst uses a different symbol to convey that the event could have been developed (broken down) further but he/she has chosen not to do so for the analysis.		An undeveloped event has the same properties as a basic event with the exception that it is graphically rendered as a diamond. The diamond representation graphically illustrates that this event could have been expanded into a separate fault tree but was not. In other words, the analyst uses a different symbol to convey that the event could have been developed (broken down) further but he/she has chosen not to do so for the analysis.

Richard House: /* XOR Gate */

2015-12-22T21:00:50Z

XOR Gate

← Older revision		Revision as of 21:00, 22 December 2015
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	===XOR Gate===		===XOR Gate===

	[[Image:16.png\|center\|100px\|\|link=]]		[[Image:16.png\|center\|100px\|link=]]

	In an XOR gate, the output event occurs if exactly one input event occurs. This is similar to an OR gate with the exception that if more than one input event occurs then the output event does not occur. For example, if there are two input events then the XOR gate indicates that the output event occurs if only one of the input events occurs but not if zero or both of these events occur. From a system reliability perspective, this would imply that a two-component system would function even if both components had failed. Furthermore, when dealing with time-varying failure distributions, and if system components do not operate through failure, a failure occurrence of both components at the exact same time ( <math>dt)\,\!</math> is an unreachable state; thus an OR gate would suffice. For these reasons, an RBD equivalent of an XOR gate is not presented here and BlockSim does not explicitly provide an XOR gate.		In an XOR gate, the output event occurs if exactly one input event occurs. This is similar to an OR gate with the exception that if more than one input event occurs then the output event does not occur. For example, if there are two input events then the XOR gate indicates that the output event occurs if only one of the input events occurs but not if zero or both of these events occur. From a system reliability perspective, this would imply that a two-component system would function even if both components had failed. Furthermore, when dealing with time-varying failure distributions, and if system components do not operate through failure, a failure occurrence of both components at the exact same time ( <math>dt)\,\!</math> is an unreachable state; thus an OR gate would suffice. For these reasons, an RBD equivalent of an XOR gate is not presented here and BlockSim does not explicitly provide an XOR gate.