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Reliability based design optimizationProbabilistic vs. deterministic designOptimal risk allocation between two failure modes.Laminate design exampleStochastic, analysis, and design surrogates.Uncertainty reduction vs. extra weight.

Deterministic design for safetyLike 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.TestsAccident investigationsRisk allocation driven by history (accidents).

Pro and cons of probabilistic designProbabilistic 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.

Optimal risk allocationIf 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?

ExampleAn 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?

Top Hat questionIn 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?ResponseCapacityBothFORM vs. Monte CarloFORM is much cheaper, butDoes 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.

.8Deterministic Design of Composite Laminates

Design of angle-ply laminateMaximum strain failure criterion

Load induced by internal pressure:NHoop = 4,800 lb./in., NAxial = 2,400 lb./in.

Physical challenge in this problemIn 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.

.10Summary of Deterministic DesignOptimal ply-angles are 27 from hoop direction

Laminate thickness is 0.1 inch

Probability of failure (510-4) is high with safety factor 1.4.

Top Hat problemIf the 27o design was built with 26o because of manufacturing reliability that wouldIncrease the chance of failure due to hoop stressIncrease the chance of failure due to axial stressIncrease the chance of failure due to matrix crackingAll of the above.

.12Reliability-based Laminate DesignFirst ply failure principle

Pt = 10-44 Design Variables1, 2, t1, t2

12 Normal Random VariablesTzero (CV = 0.03)1, 2 (CV = 0.035)E1, E2, G12, 12 (CV = 0.035)1c, 1t (CV = 0.06)2c, 2t, 12u (CV = 0.09)

xueyong@mae.ufl.edu13Response Surface OptionsDesign response surface approximation (DRS)Response or Probability v.s. design variables: G=G(d)Used in optimizationStochastic 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 optimizationAnalysis response surfaces Response v.s. random variables + design variables: G=G(x, d)Advantage: improve efficiency of SRSChallenge:Construct RS in high dimensional space ( > 10 variables)

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu14Analysis Response Surfaces (ARS)Fit strains in terms of 12 variablesDesign of experiments:Latin Hypercube Sampling (LHS)D.V.R.V.StrainARS

Probabilities calculated by MCS based on fitted polynomialsReduce computational cost of MCS

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu15Reliability-based Design OptimizationDesign Response Surface (DRS)Fit to Probability in terms of 4 D.V.Filter out noise generated by MCSUsed in RBDOitiProbability

ARSDOE & MCSDRSOptimizationConverge?StopYesNoNo

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu16Approximation

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu17OptimizationDeterministic, Reliability-based, and Simplified designs

The thickness is high for application

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu18Improving Reliability-based DesignReliability-based designThickness of 0.12 inchProbability of failure of 10-4 levelMust reduce uncertainties:Quality control (QC)Reject small numbers of poor specimenTruncate distribution of allowables at lower side (2 )Reduce material scatterReduce Coefficient of Variation (CV)Better manufacture process (Better curing process)Improve allowablesIncrease Mean Value of allowablesNew materials

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu19Change Distribution of 2 allowableReduce scatter (CV) by 10%Increase allowable (Mean value) by 10%

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu20Quality Control (QC) on 2 allowableReduce probability of failureReduce thickness

Structural & Multidisciplinary Optimization Groupxueyong@mae.ufl.edu21Tradeoff PlotTo be chosen by the cost of implementing these methods

Structural & Multidisciplinary Optimization Group

NAxial

NHoop

x

y

1

2

(

_1027161103.unknown

Strain = g(1, 2, t1, t2, E1, E2, G12, (12, (1, (2, Tzero, Tservice)POF = p(1, 2, t1, t2)Design variables(1(2 t1 (inch)t2 (inch)

Range20( to 30(20( to 30(0.0125 to 0.030.0125 to 0.03

DRS Error StatisticsFCCCD 25 pointsLHS 252 points

quadratic5th order

Rsquare Adj.0.6860.998

Standard error (probability)5.3e-40.12e-4

Mean of Response (probability)3.2e-40.44e-4

ARS Error StatisticsQuadratic ARS based on LHS 182 points

(2 in (1

Rsquare Adj.0.996

Standard error (millistrain)0.060

Mean of Response (millistrain)8.322

Ply AnglesThickness (inch)Probability of Failure

Deterministic[((27.0)2/((27.0)3]S0.105e-4

Reliability[((24.9)3/((25.2)3]S0.120.55e-4

Simplified[((25)6]S0.120.57e-4

(0.12 inch)Probability of failure

CV = 0.09CV = 0.081

[((25)6]S0.57e-40.11e-4

(0.12 inch)Probability of failure

E(2u) = 0.0154E(2u) = 0.01694

[((25)6]S0.57e-40.03e-4

(0.12 inch)Probability of failure

NormalTruncate at -3((14 out of 10,000)Truncate at -2(

(23 out of 1,000)

[((25)6]S0.57e-40.001e-4< 1e-7

(POF=1e-4)0.12 inch0.10 inch0.08 inch

NormalTruncate at 2.8( (26 out of 10,000)Truncate at -1.35( (90 out of 1,000)

[((25)6]S0.57e-40.9e-41e-4

Chart10.1734290.0996120.0197130.0970040.0005980.00010.0285280.0089490.003130.0176150.000000020.00010.005440.0000410.0004850.0023150.0010640.00000010.0000680.0003240.0001920.110.00001150.000050.0000510.00000230.0000110.0000070.00000050.00000060.0000012

NominalQuality control to -2 Sigma10% increase in allowable10% reduction in variabilityAllThickness (inch)Failure Probability

plot_untrunNominalTruncate Epsilon2_u at -2 Sigma1.1*E(Epsilon2_u)0.9*COV(Epsilon2_u)Alltheta1theta2t1t2tpofpofpofpofpof25250.008750.008750.070.1734299.96E-021.97E-029.70E-025.98E-0425250.010.010.082.85E-028.95E-033.13E-031.76E-022.00E-0825250.011250.011250.095.44E-034.10E-054.85E-042.32E-0325250.01250.01250.11.06E-031.00E-076.80E-053.24E-0425250.013750.013750.111.92E-041.15E-055.00E-0525250.0150.0150.125.10E-052.30E-061.10E-0525250.0150.020.147.00E-065.00E-076.00E-0725250.020.020.161.20E-060.061.00E-040.161.00E-04

plot_untrun0.1734290.0996120.0197130.0970040.0285280.0089490.003130.0176150.005440.0000410.0004850.0023150.0010640.00000010.0000680.0003240.0001920.110.00001150.000050.0000510.00000230.0000110.0000070.00000050.00000060.0000012

POFPOF(Epsilon2_u at 2 Sigma)POF(1.1*E(Epsilon2_u))POF(0.9*COV(Epsilon2_u))Thickness (inch)

0.1734290.0996120.0197130.0970040.0005980.0285280.0089490.003130.0176150.000000020.005440.0000410.0004850.0023150.0010640.00000010.0000680.0003240.0001920.110.00001150.000050.0000510.00000230.0000110.0000070.00000050.00000060.0000012

NominalQuality control to -2 Sigma10% increase in allowable10% reduction in variabilityAllThickness (inch)Failure Probability