Reliability based design optimization Probabilistic vs. deterministic design – Optimal risk...
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Reliability based design optimization • Probabilistic vs. deterministic design – Optimal risk allocation between two failure modes. • Laminate design example – Stochastic, analysis, and design surrogates. – Uncertainty reduction vs. extra weight.
Reliability based design optimization Probabilistic vs. deterministic design – Optimal risk allocation between two failure modes. Laminate design example
Reliability based design optimization Probabilistic vs.
deterministic design Optimal risk allocation between two failure
modes. Laminate design example Stochastic, analysis, and design
surrogates. Uncertainty reduction vs. extra weight.
Slide 3
Deterministic design for safety Like probabilistic design it
needs to lead to low probability of failure. Instead of calculating
probabilities of failure use array of conservative measures. Safety
factors. Conservative material properties. Tests Accident
investigations Risk allocation driven by history (accidents).
Slide 4
Pro and cons of probabilistic design Probabilistic design
requires more data, that is often not available or expensive to
get. Probabilistic design may require to accept finite probability
of death or injury and may lead to legal liabilities. Probabilistic
design may allow more economical risk allocation. Probabilistic
design may allow trading measures for compensating against
uncertainty against measures for reducing it.
Slide 5
Optimal risk allocation If there is a single failure mode, the
chances are that history has resulted in safety factors that
reflect the desired probability of failure. When there are multiple
failure modes it makes sense to have excessive protection against
modes that are cheap to protect against. Adding probabilities : If
one mode has failure probability p1 and a second p2, what is the
system failure probability if they are independent?
Slide 6
Example An airplane wing weighs 10,000 lb and the tail weighs
1,000 lb. With a safety factor of 1.5, each has a failure
probability of 1%, for a total failure probability of 2% (actually
1-0.99^2) For each component the relation between the probability
of failure and additional weight is Reduce the failure probability
to 0.5% with minimum weight. Adding 200 lb to wing and 20 lb to
tail reduces the probabilities of each by a factor of 4 for 220
lbs. Adding 120 lb to the wing and 80 lb to the tail will lead to
0.435% wing failure probability plus 0.004% tail failure
probability. Safer and lighter. What is the optimum?
Slide 7
Top Hat question In a design problem g=r-c, and the costs of
changing the means of r or c by one unit are the same. The standard
deviation of r is twice that of c. Which mean should we change to
reduce the Pf at minimum cost? Response Capacity Both
Slide 8
FORM vs. Monte Carlo FORM is much cheaper, but Does not give
you good estimate of system probability of failure when failure
modes are strongly coupled. Can have large errors when variables
are far from normal and limit state have multiple local MPPs. More
difficult to allocate risk. MCS usually too expensive unless you
fit a surrogate to limit state function.
Slide 9
. 8 Deterministic Design of Composite Laminates Design of
angle-ply laminate Maximum strain failure criterion Load induced by
internal pressure: N Hoop = 4,800 lb./in., N Axial = 2,400
lb./in.
Slide 10
Physical challenge in this problem In a cylinder under internal
pressure the stresses in the hoop directions are twice those in the
axial direction, and so you could put fibers in both directions,
but twice as many in the hoop direction. However, fibers shrink
much less than matrix at low temperatures, so the fibers in hoop
direction will not allow the matrix of the axial fibers to shrink,
causing it to crack. We have to compromise by having fibers in
intermediate directions with less than 90-degrees between fibers in
different layers of laminate.
Slide 11
. 10 Summary of Deterministic Design Optimal ply-angles are 27
from hoop direction Laminate thickness is 0.1 inch Probability of
failure (5 10 -4 ) is high with safety factor 1.4.
Slide 12
Top Hat problem If the 27 o design was built with 26 o because
of manufacturing reliability that would Increase the chance of
failure due to hoop stress Increase the chance of failure due to
axial stress Increase the chance of failure due to matrix cracking
All of the above.
Slide 13
. 12 Reliability-based Laminate Design First ply failure
principle P t = 10 -4 4 Design Variables 1, 2, t 1, t 2 12 Normal
Random Variables T zero (CV = 0.03) 1, 2 (CV = 0.035) E 1, E 2, G
12, 12 (CV = 0.035) 1 c, 1 t (CV = 0.06) 2 c, 2 t, 12 u (CV =
0.09)
Slide 14
Structural & Multidisciplinary Optimization Group
[email protected] 13 Response Surface Options Design response
surface approximation (DRS) Response or Probability v.s. design
variables: G=G(d) Used in optimization Stochastic response surface
approximation (SRS) Response v.s. random variables: G=G(x) Used in
probability calculation. Need to construct SRS at every point
encountered in optimization Analysis response surfaces Response
v.s. random variables + design variables: G=G(x, d) Advantage:
improve efficiency of SRS Challenge: Construct RS in high
dimensional space ( > 10 variables)
Slide 15
Structural & Multidisciplinary Optimization Group
[email protected] 14 Analysis Response Surfaces (ARS) Fit strains
in terms of 12 variables Design of experiments: Latin Hypercube
Sampling (LHS) D.V. R.V. Strain ARS Probabilities calculated by MCS
based on fitted polynomials Reduce computational cost of MCS
Slide 16
Structural & Multidisciplinary Optimization Group
[email protected] 15 Reliability-based Design Optimization Design
Response Surface (DRS) Fit to Probability in terms of 4 D.V. Filter
out noise generated by MCS Used in RBDO ii titi Probability ARS DOE
& MCS DRS Optimization Converge? Stop Yes No
Slide 17
Structural & Multidisciplinary Optimization Group
[email protected] 16 Approximation
Slide 18
Structural & Multidisciplinary Optimization Group
[email protected] 17 Optimization Deterministic,
Reliability-based, and Simplified designs The thickness is high for
application
Slide 19
Structural & Multidisciplinary Optimization Group
[email protected] 18 Improving Reliability-based Design
Reliability-based design Thickness of 0.12 inch Probability of
failure of 10 -4 level Must reduce uncertainties: Quality control
(QC) Reject small numbers of poor specimen Truncate distribution of
allowables at lower side (2 ) Reduce material scatter Reduce
Coefficient of Variation (CV) Better manufacture process (Better
curing process) Improve allowables Increase Mean Value of
allowables New materials
Slide 20
Structural & Multidisciplinary Optimization Group
[email protected] 19 Change Distribution of 2 allowable Reduce
scatter (CV) by 10% Increase allowable (Mean value) by 10%
Slide 21
Structural & Multidisciplinary Optimization Group
[email protected] 20 Quality Control (QC) on 2 allowable Reduce
probability of failure Reduce thickness
Slide 22
Structural & Multidisciplinary Optimization Group
[email protected] 21 Tradeoff Plot To be chosen by the cost of
implementing these methods