Transcript Introduction to structural uncertainties

Uncertainty and Safety Measures
How do we classify uncertainties? What are their
sources?
– Lack of knowledge vs. variability.
What type of safety measures do we take?
– Design, manufacturing, operations & postmortems
– Living with uncertainties vs. changing them
How do we represent random variables?
– Probability distributions and moments
Reading assignment
Oberkmapf et al. “Error and uncertainty in modeling and simulation”,
Reliability Engineering and System Safety, 75, 333-357, 2002
S-K Choi, RV Grandhi, and RA Canfield, Reliability-based structural design,
Springer 2007. Chapter 1 and Section 2.1.
Available on-line from UF library
http://www.springerlink.com/content/w62672/#section=3200
07&page=1
Source: www.library.veryhelpful.co.uk/ Page11.htm
Modeling uncertainty (Oberkampf
et al.)
.
Classification
of uncertainties
Aleatory uncertainty:
Inherent variability
– Example: What does
regular unleaded cost in
Gainesville today?
Epistemic uncertainty
Lack of knowledge
Source: http://www.ucan.org/News/UnionTrib/
– Example: What will be the average cost of regular unleaded next
January 1?
Distinction is not absolute
Knowledge often reduces variability
– Example: Gas station A averages 5 cents more than city average
while Gas station B – 2 cents less. Scatter reduced when
measured from station average!
British Airways 737-400
A slightly different
uncertainty
classification
.
Distinction between Acknowledged and Unacknowledged errors
Safety measures
Design: Conservative loads and material
properties, accurate models, certification
of design
Manufacture: Quality control, oversight
by regulatory agency
Operation: Licensing of operators,
maintenance and inspections
Post-mortem: Accident investigations
Many players invest to reduce uncertainty in
aircraft structures.
The federal government (e.g. NASA)
develops more accurate models and measurement
techniques. A or E?
Boeing performs higher fidelity simulations and
high accuracy manufacturing. A or E?
Airlines invest in maintenance and inspections. A or E?
FAA certifies aircraft & pilots. A or E?
NTSB, FAA and NASA fund accident investigations.
oblems uncertainty
1. List at least six safety
measures or uncertainty
reduction mechanisms
used to reduce highway
fatalities of automobile
drivers.
Source: Smithsonian Institution
Number: 2004-57325
2. Give examples of
aleatory and epistemic uncertainty faced by
car designers who want to ensure the safety of
drivers.
Representation of
uncertainty
Random variables: Variables that can take
multiple values with probability assigned to
each value
Representation of random variables
– Probability density function (PDF)
– Cumulative distribution function (CDF)
– Moments: Mean, variance, standard
deviation, coefficient of variance (COV)
Probability density function (PDF)
• If the variable is discrete, the
probabilities of each value is the
probability mass function.
• For example, with a single die,
toss, the probability of getting 6 is
1/6.If you toss a pair of dice the
probability of getting twelve (two
sixes) is 1/36, while the
probability of getting 3 is 1/18.
• The PDF is for continuous
variables. Its integral over a
range is the probability of being in
that range.
Histograms
• Probability density functions have to be inferred from
finite samples. First step is a histogram.
• Histograms divide samples to finite number of ranges
and show how many samples in each range (box)
• Histograms below generated from normal distribution
with 50 and 500,000 samples.
5
2
x 10
1.8
14
1.6
12
1.4
10
1.2
1
8
0.8
6
0.6
4
0.4
0.2
2
0
0
7.5
8
8.5
9
9.5
10
10.5
11
11.5
12
5
6
7
8
9
10
11
12
13
14
15
Number of boxes
• Each time we sample, histogram will be different.
• Standard deviation (from sample to sample) of the height
n of a box is approximately 𝑛. Keep that below change
in n from one box to next.
• Histograms below were generated with 5,000 samples
from normal distribution.
• With 8 boxes s.d. relatively small (~25) but picture is
coarse. With 20 boxes it’s about right. With 50, s.d. is too
high (~10) relative to change from one box to next.
350
800
1800
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1600
1400
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500
1000
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800
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600
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50
100
200
0
-4
-3
-2
-1
0
1
2
3
4
0
-4
-3
-2
-1
0
1
2
3
4
0
-4
-3
-2
-1
0
1
2
3
4
Histograms and PDF
How do you estimate the PDF from a histogram?
Only need to scale.
P robability distribution function f :
P(a  x  a  da)  f ( x)da
Cumulative distribution function
x
F ( x)  P( X  x) 
Integral of PDF

f (t )dt

1
Experimental CDF
from 500 samples
shown in blue,
compares well to
exact CDF for
normal
distribution.
0.9
0.8
normal CDF
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-4
-3
-2
-1
0
x
1
2
3
4
Problems CDF
• Our random variable is the number seen
when we roll one die. What is the CDF of
2?
• Our random variable is the sum on a pair
of dice. What is the CDF of 2? Of 13?
Probability plot
• A more powerful way to compare data to a
possible CDF is via a probability plot (500
points here)
Probability plot for Normal distribution
0.9999
0.9995
0.999
0.995
0.99
Probability
0.95
0.9
0.75
0.5
0.25
0.1
0.05
0.01
0.005
0.001
0.0005
0.0001
-4
-3
-2
-1
0
Data
1
2
3
4
Moments
• Mean
• Variance
 ( X )   xf ( x)dx  E[ X ]
2
Var ( X )   ( x   ) 2 f ( x)dx  E  X    


• Standard deviation
• Coefficient of variation
• Skewness
 X   3 
E 
 

 

  Var ( X )

COV 
