lecture 3: probabilistic designisdl.cau.ac.kr/education.data/complex.sys/lecture 3.pdfcantilever...
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Lecture 3: Probabilistic Design
Uncertainty in Engineering Systems and Risk Managements
Professor CHOI Hae-Jin
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Contents
• Error Propagation
• Decision-making under Uncertainty
• Probabilistic Design
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Conventional Approach
• Conventional engineering design uses a deterministic approach. It disregards the fact that material properties, the dimensions of the parts, and the externally applied loads vary statistically.
• In conventional design, theses uncertainties are handled by applying a factor of safety
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Probabilistic Approach
• In critical design situation, such as aircraft, space, and nuclear applications, it is often necessary to use a probabilistic approach for quantifying uncertainty and increasing reliability of a system.
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Error Propagation
• When working with random variables, it is necessary to propagate error (variability) through systemic equations (or models).
• For example, we need to know the variability of the deflection of the cantilever beam with a given variability of the load
3
3
L
EI
P
???δP
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Error Propagation
• For normal distributions, following equations are a method for estimating propagated errors.
1 2
1/ 22
2
1
when an output is ( ),
the mean of is ( , ,..., ), and
the standard deviation of is
ny x x x
n
y xi
i ix
1 2 ny x , x , ..., x
y
y (5.6)
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Error Propagation
• The estimate of the mean of a function relationship comes from substituting the mean values of the random variables.
• The estimate of the variance of a function relationship is simply the weighted variances of the constitutive variances, the weighting factors being the squares of the partial derivatives evaluated at the means
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Mean and standard deviations for simple algebraic operations (x, y : independent
random variables)
Function Mean Standard Deviation
a a 0
x x x
ax x a x
ax xa xa
x y x y
1/ 22 2
x y
x y x y
1/ 22 2
x y
xy x y 2 2 2 2 1/ 2( )x y x y x yC C C C
/x y /x y 1/ 2
2 2 2
/ / 1x y x y yC C C
1/ x 211 x
x
C
21xx
x
CC
x 211
8x xC
21
12 16
x
x xC C
nx 2( 1)
12
n
x x
n nC
22( 1)
14
n
x x x
nn C C
Where, Cx =σx
μx Coefficient of variation of
the random variable. x
Table 5.3
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Error Propagation by Simulation
• In following situation, it is very difficult to estimate error propagation by the method.
– Distribution of random input is not a normal distribution (such as, lognormal and Weibull)
– Function is not a form of mathematics, but computer simulation or experiments.
• We can employ Monte Carlo simulation for propagating the error (variability).
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Error Propagation by Simulation
• Procedure of the Monte Carlo simulation1. Define a domain of possible inputs.
2. Generate an instance of inputs randomly from the domain using random number generator.
3. Perform a deterministic computation using the instance.
4. Repeat step 2 and 3 to collect enough amount of data
5. Aggregate the results of the individual computations into the final result to estimate a statistical distribution
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Example of Error Propagation
• Example 6.1: the load of the cantilever beam varies as P ~ N(100, 10) N, what are the mean and standard deviation of deflection?
Deterministic parametersE = 200 GPaI = 1000 mm4
L = 500 mm
P
L
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Example of Error Propagation
• δ= (L3/3EI)P = AP where
• Deterministic parameter
• A=L3/3EI =
• µP = 100 and σP2 = 10
• From the Table 5.3 or Eq. 5.6, µδ = AµP and σδ=AσP
• µδ = AµP = 0.20833 x 100 = 20.833 (mm)
• σδ= AσP = 0.20833 x 3.16 = 0.6587 (mm)
-3 3 3
9 12
(500 10 ) 5 10(m/N)
3 200 10 1000 10 24
0.20833 (mm/N)
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Probabilistic Decision-making
• Example 6.2: The requirement of the cantilever beam deflection is less than 22 (mm). Does this beam design satisfy the requirement with 99% chance?
Find the deflection limit (critical point) of 99% percentile.
0.99
δ=??µδ = 20.833 (mm)
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Probabilistic Decision-making
lim,0.99 lim,0.99
lim,0.99
x=2.326 at =0.01 (i.e.,1- 0.99)
20.833x= =2.326
0.6587
20.833 2.326 0.6587 22.37 22
From Table 5.2, the critical point is
or
Therefore,
This beam design is NOT acceptable. 0.99
δlim,0.99µδ = 20.833 (mm)
x
1-Ф(x)=α
=2.326
=0.010.99
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Probabilistic Design Approach
• Procedure of probabilistic design approach
1. Identify sources of uncertainty and system constraints (such as yield strength, deflections, etc.),
2. Establish system function (model),
3. Categorize system parameters: random variables, deterministic parameters, and design variables,
4. Find the distribution of system output by error propagation, and
5. Determine the values of design variables
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Probabilistic Design Approach
• Example 6.3: the load of the cantilever beam varies as P ~ N(100, 10) N, determine the maximum length of beam so that the deflection of the beam is less than 22(mm) with 99% chance.
P
L
Deterministic parametersE = 200 GPaI = 1000 mm4
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Probabilistic Design Approach
• STEP 1: Identify sources of uncertainty and system constraints
• Source of uncertainty is in loading, P ~ N(100,10) N
• System constraints: Deflection < 22 (mm)
• STEP 2: Establish system function (model)
• Deflection δ = PL3/(3EI)
• STEP 3: Categorize system parameters
• Random variables: P
• Deterministic parameters: E, I
• Design variable: L
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Probabilistic Design Approach
• STEP 4: Find the distribution of output by error propagation
δ= (L3/3EI)P = (1/3EI) PL3 = A PL3
Where the deterministic parameters, Α =1/3EI =
1/(3*200*10e9*1000*10e-12) = 1/600
Mean of propagated error, µδ = A µP L3
Standard deviation of propagated error, σδ = A σP L3
Therefore, the estimated distribution of deflection
δ ~ N(µδ, σδ2)
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Probabilistic Design Approach
• STEP 5:Determine the values of design variables
3
lim,0.99lim,0.99
3
3 3 -3
lim,0.99
-3 -3
3
x=2.326 at =0.01 (i.e.,1- 0.99)
x= =2.326
2.326 22 10
22 10 22 10L
A( 2.326 ) (1/
p
p
p p
p p
A L
A L
or A L A L
From Table 5.2 the critical point is
Therefore,
3 0.4973600)(100 2.326 3.162)
497.3( )mm