Fuzzy Fault Tree Analysis of Deep Foundation Pit Supporting Systems Failure Probability Y'H'BIAN Dep

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Fuzzy Fault Tree Analysis of Deep Foundation Pit
SupportingSystems Failure ProbabilityY.H.BIAN
Department of Geotechnical Engineering of Tongji
University, Shanghai2004-11-6
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outline

• 1.Introduction
• 2.Fuzzy set theory
• 3.Fuzzy probability of deep foundation pit
  calculated by fault tree
• 4.Sensibility analysis
• 5.Conclusion

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1.Introduction

• Cause of risk analysis theory introduced in deep
  foundation pit
• Accidents occurred in deep foundation have leaded
  to huge economical losses ,casualties, work stage
  delay, bad social effect
• Risk analysis can reduce the failure probability
  and losses of deep foundation pit

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• Calculation of failure probability of deep
  foundation pit supporting systems using fault
  tree
• Limitation of conventional fault tree analysis
• Ignore the data uncertainty
• Fuzzy set used in fault tree analysis
• Fuzzy failure probability replace failure
  probability of basic events
• Triangular fuzzy number represent failure
  probability of basic events

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2.Fuzzy set theory

• 2.1Basic concepts of fuzzy value
• A fuzzy set initiated by Zadeh in
  1965 is defined as follows
• Let X denote a universal set. Then
  a fuzzy subset A of X is defined by its
  membership function.
• A fuzzy subset A can be characterized
  as a set of ordered pairs of element x and its
  grade , and is often written as
• L-R fuzzy number with only one
  point which membership grade is 1 is widely used
  in practical project , the membership function is

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A linear LR fuzzy number (triangular membership
function) is used in case of lack of adequate
data. The membership function is
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2.2Fuzzy number operation
For given in the the internal ,
according to the extension principle proposed by
Zadeh, the arithmetic operations of fuzzy numbers
can be defined by means of -cut operations
Then
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3.Fuzzy failure probability of deep foundation
pit calculated by fuzzy fault tree
3.1 Work out a fault tree of deep foundation pit
supporting system
Fault tree of diaphragm wall supporting system in
deep foundation pit is shown in Figs.1 and
explained as follows.
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3.2 Calcualtion of accident probability
Fault tree in Figs.1 is obtained from Boolean
algebra as follows
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From the result obtained above, top event
is the union of 25 sets which are the minimum cut
set of the fault tree, i.e., X1X17,X1X18,X1X1
9,X1X20,X1X26,X1X27,X1X28X29,X2X8,X2X1
9,X2X21,X2X22,X2X23,X2X24,X6X7,X3X20,
X3X21,X3X25,X3X26,X3X27,X3X28X29,X25,
X24,X10,X11,X4X12X13, which are
corresponding to 25 accident modes.
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In practical engineering, occurrence
probability of top event (PT) is obtained using
approximate probability formula of independent
events as shown in Eqs.(8)
4.Sensibility analysis
According to the fuzzy number model defined in
this paper, sensibility evaluation index is
simply defined as shown in Eqs.(9).
(9)
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5.Conclusion
If ViVj, which shows that it is more
effective to reduce occurrence probability of top
event by reducing occurrence probability of event
i than event j.

• Fuzzy fault tree analysis can not only achieve
  the analytical objective of classical
  probability, but also obtain possible failure
  probability distribution of deep foundation pit
  supporting system by integrating a small amount
  statistical data and experience of in-situ
  engineer.

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• The same membership function(triangular
  membership function ) is introduced in occurrence
  probability of all basic events. Different
  membership functions can be used in different
  basic events in practical consideration.

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Thank you!