A COMPUTATIONAL MODEL FOR AIRCRAFT STRUCTURAL RELIABILITY UPDATING AND INSPECTION OPTIMIZATION USING

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A COMPUTATIONAL MODEL FOR AIRCRAFT STRUCTURAL
RELIABILITY UPDATING AND INSPECTION OPTIMIZATION
USING BAYESIAN UPDATING
Amit Kale Department of Mechanical and Aerospace
Engineering University of Florida.
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Motivation

• Aircraft structures are designed for very low
  probability of structural failure, however
  structural failure is inevitable.
• Uncertainty in material properties (strength)
• Defects and flaws present in the structure.
• Environmental factors
• If uncertainty can be reduced, structural
  reliability can be improved.
• Develop a method to integrate new information
  with existing data to reduce uncertainty
• Risk informed decision and corrective actions can
  be taken to prevent structural failure

Aloha Airlines Flight 243 Accident, 1988
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Objectives

• Develop a computational method to use in-service
  observed data from inspections to obtain new
  estimates of material properties.
• Monte Carlo simulation
• Bayesian updating
• Use these new estimates to modify inspections and
  maintenance plans to prevent structural failure.
• Formulate an optimization problem to determine
  the number of inspections required to obtain a
  specified confidence in the estimated data
• Confidence level specified by CVaR

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Structural failure

• Failure due to loss of structural strength beyond
  what is required to carry service loads
• Loads exceeding the design limit of the
  structure.
• Crack and growth due to
• Fabrication defects (ai)
• Applied loading (s)
• Environmental factors like corrosion
• Stochastic inputs
• ai , lognormally distributed
• s, lognormally distributed
• m, lognormally distributed

ai
ac
aN
crack
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Bayesian updating for structural inspections

• Bayesian updating is extensively used to
  integrate initial information with observed data
  from inspections
• Prior is represented by A and the observed
  information is represented by B.
• Advantages
• Filters out anomaly, errors in data.
• Obtains accurate estimate of random variables.

• In this paper, observed crack length (aN B) is
  used to update prior distribution of material
  property (A m)

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Basic reliability computation Calculating next
inspection time
Specify structural design and required
reliability level (Pfth)
Conduct reliability analysis and crack growth
analysis to obtain the probability of damage
exceeding unsafe level during service life.
Schedule inspection when Pfth is exceeded.
Replace the detected damage with new components
Compute failure probability Pf at end of service
life.
Is Pf gt Pfth
Stop
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Bayesian updating at each inspection using Monte
Carlo
Conduct inspections to detect cracks at optimum
times
Inspect a structural component. If a crack is
detected (Ad) use Eq. 1 else use Eq. 2 to obtain
new Estimates of m
No
Check if the m has been estimated with specified
confidence
Count number of inspected components
Yes
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Results

• Inspection times obtained using only the prior
  information on m for maintaining a probability of
  failure (crack size exceeding critical size) of
• 10-7
• At the first inspection (17671 f lights) ,
  structural components are inspected and the
  observed crack length are used to update material
  property m. Inspection times are updated using
  the updated m

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No-CVaR constraint

• 200 structural components are inspected, and no
  constraint on accuracy of updated parameter is
  placed.
• One additional inspection is needed if updated
  parameter m is used

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80-CVaR constraint

• Number of inspected components for MCS is
  determined from CVaR constraint
• Inspection times and number of inspection
  required has error because of inaccurate
  estimation of m

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70-CVaR constraint

• Number of inspected components for MCS is
  determined from CVaR constraint
• Inspection times and number of inspection
  required has error because of inaccurate
  estimation of m
• Large MCS sample size is required for accurate
  estimation using Bayesian updating.

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Proof of test for Bayesian updating

• To demonstrate that the prior assumed
  distribution of m when updated sufficient number
  of times will converge to the true distribution
• We assume that true m-2 is deterministic, 0.5
• Assumed distribution lognormal, mean 0.9, std
  0.5

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Conclusions and Future work

• Observed data from inspections can be used to
  update random variables and obtain refined
  estimate of structural reliability using Bayesian
  analysis.
• Use of CVaR constraints on updated parameters can
  help reduce computational cost while maintaining
  desired accuracy.
• Future work will demonstrate the extension of
  this methodology to updating multiple random
  variables from single observed data.

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FORM First Order Reliability Method

• Convert all random variables (xs) to standard
  normal variables (us).
• Convert the failure function g(x) in design
  space to standard normal space G(u) (limit
  state).
• Minimize distance from origin to the limit state
  (ß).

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Monte Carlo Simulation

• Generate N set of random variables (xs).
• Evaluate the failure function for each set and
  check if it lies in the failed region.
• Count total number of failures Nf