Why need probabilistic approach?

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Why need probabilistic approach?
? ?

• Rain probability
• How does that affect our behaviour?

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Uncertainties in Engineering

• Natural Hazards
• Material Properties
• Design Models
• Construction Errors

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Absolute Safety Not Guaranteed

• Engineers need to
• model, analyze, update uncertainties
• evaluate probability of failure

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Questions

• What is acceptable failure probability?
• - stadium vs shed

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Questions

• Should one want to be conservative if a perfectly
  safe system is possible?
• - overbooking in airlines
• - parking permits

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Questions

• Should one minimize risk if money is not a
  problem?
• - system consideration
• e.g.dam

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Trade-off Decision Analysis

• Risk vs. consequence
• System risk

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Formal analysis of uncertainties and probability

• Not all problems can be solved by analysis of
  data
• Set Theory
• Sample space collection of all possibilities
• Sample point each possibility
• Event subset of sample space
• Probability Theory

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Union either E1 or E2 occur E1?E2
Intersection both E1 and E2 occur E1n E2
or E1 E2
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Examples
B
A
No communication between A and B E1E2
No communication between A and B E3?E1E2
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Example - pair of footings
Settlement occurs E1?E2
Tilting occurs E1E2?E1E2
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de Morgans rule
Water Supply
2
1

• E1 pipe 1 breaks
• E2 pipe 2 breaks

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Event of no failure
Extension to n events
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de Morgan 2
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Basis of Probability Estimation

1. Subjective assumption e.g. P(Q) 1/2
2. Relative frequency e.g. P(Q)502/1000
3. Bayesian (a)(b)

judgment limited observation
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Probability of Union
in general
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Using de Morgans rule
conditional probability
P (intersection)
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or
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Statistical independence
s.i.

• if E1 and E2 are s.i.

or
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Example
P(E1) 0.1 P(E2)0.1 P(E3) 0.1
E1 and E2 are not s.i.
E1 and E3 are s.i.
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if E1 and E2 are s.i.
if all are s.i.
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s.i. and m.e. (mutually exclusive)
if E1 and E2 are m.e.
if E1 and E2 are s.i.
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E1 ? is open
P 2.15
P(E1)2/5
P(E2)3/4
P(E3)2/3
P(E3E2)4/5
P(E1E2E3)1/2
a) P(go from A to B through C)
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b)
P(go from A to B)
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T.O.T (Theorem of Total Probabilities)
P(A) P(AE1)P(E1)P(AE2)P(E2)P(AEn)
P(En)
Eis are m.e. and c.e.
Bayes theorem
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E 2.30
aggregate for construction
engineer's judgment based on geology and
experience
crude test
reliability (or quality) is as follows
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After 1 successful test, what is P(G)?
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After another successful independent test,
P(G)?
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What if the two tests were performed at the same
time?
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Random variables

• A device to
• formalize description of event
• facilitate computation of probability

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CDF
PX(x)
FX(x)
PMF
fX(x)
FX(x)
PDF
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Main descriptors of R.V.

• The PMF or PDF completely define the r.v.
• Descriptors give partial information about the
  r.v.

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Mean value
Define ? E(X)
expected value of X or mean value of X
a measure of central tendency
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Measure of spread

• Standard deviation ?X
• ?X dimensionless
  
• range

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Expected value of function
recall
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Recall
After some algebra,
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Normal distribution
X N (?, ?)
N (5, 2)
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Effect of varying parameters (? ?)
fX(x)
? ? for C ? ? for B
B
C
A
x
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Standard normal distribution
S N (0,1)
fX(x)
x
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Page 380 Table of Standard Normal Probability
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Example retaining wall
Suppose X N(200,30)
x
F
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If the retaining wall is designed such that the
reliability against sliding is 99, How much
friction should be provided?
2.33
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Lognormal distribution
Parameter l ?
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Parameters ? ?
for ? ? 0.3,
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Probability for Log-normal distribution
If a is xm, then ? is not needed.
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Other distributions

• Exponential distribution
• Triangular distribution
• Uniform distribution
• Rayleigh distribution

p.224-225 table of common distribution
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Exponential distribution
x ? 0
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Beta distribution
q 2.0 r 6.0
probability
b 12
a 2.0
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Standard beta PDF
(a 0, b 1)
fX(x)
q 1.0 r 4.0
q r 3.0
q 4.0 r 2.0
q r 1.0
x
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Bernoulli sequence
S
F
p probability of a success

• Discrete repeated trials
• 2 outcomes for each trial
• s.i. between trials
• Probability of occurrence same for all trials

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Binomial distribution
S
F
x number of success
p probability of a success
P ( x success in n trials) P ( X x n, p)
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Examples

• Number of flooded years
• Number of failed specimens
• Number of polluted days

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Example
Given probability of flood each year 0.1
Over a 5 year period
P ( at most 1 flood year) P (X 0) P(X1)
0.95 0.328 0.919
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P (flooding during 5 years) P (X ? 1) 1 P(
X 0) 1- 0.95 0.41
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For Bernoulli sequence Model

• No. of success ? binomial distribution
• Time to first success ? geometric distribution
• E(T) 1/p return period

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Significance of return period in design
Service life
Suppose a bldg is expected to last 100 years and
if it is designed against 100 year-wind of 68.6
m/s
design return period
P (exceedence of 68.6 m/s each year) 1/100
0.01
P (exceedence of 68.6 m/s in 100th year) 0.01
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P (1st exceedence of 68.6 m/s in 100th year)
0.9999?0.01 0.0037
P (no exceedence of 68.6 m/s within a service
life of 100 years) 0.99100 0.366
P (no exceedence of 68.6 m/s within the return
period of design) 0.366
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If it is designed against a 200 year-wind of 70.6
m/s
P (exceedence of 70.6 m/s each year) 1/200
0.005
P (1st exceedence of 70.6 m/s in 100th year)
0.99599?0.005 0.003
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P (no exceedence of 70.6 m/s within a service
life of 100 years) 0.995100 0.606 gt 0.366
P (no exceedence of 70.6 m/s within return period
of design) 0.995200 0.367
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How to determine the design wind speed for a
given return period?

• Get histogram of annual max. wind velocity
• Fit probability model
• Calculate wind speed for a design return period

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Frequency
Example
Annual max wind velocity
Design for return period of 100 years p 1/100
0.01
?V100 90.6 mph
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Alternative design criteria 1
Suppose we design it for 100 mph, what is the
corresponding return period?
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Probability of failure
Pf P (exceedence within 100 years) 1- P (no
exceedence within 100 years) 1- (1-0.000233)100
0.023
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P ( x occurrences in n trials)
x 0, 1, 2,
Poisson distribution
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P 3.42
Service stations along highway are located
according to a Poisson process Average of 1
station in 10 miles ? n 0.1 /mile
P(no gasoline available in a service station)
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No. of service stations
(a)
P( X ? 1 in 15 miles ) ?
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(b) P( none of the next 3 stations have
gasoline)
No. of stations with gasoline
binomial
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(c) A driver noticed the fuel gauge reads empty
he can go another 15 miles from experience.
P (stranded on highway without gasoline) ?
No. of station in 15 miles
P (S)
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x P( S X x ) P( X x ) P( S X x ) P( X x )
0 1 e-1.5 0.223 0.223
1 0.2 1.5 e-1.5 0.335 0.067
2 0.22 1.52/2! e-1.5 0.251 0.010
3 0.23 1.53/3! e-1.5 0.126 0.001
4 0.24 1.54/4! e-1.5 0.047 0.00007
Total 0.301
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Alternative approach
Mean rate of service station 0.1 per
mile Probability of gas at a station 0.8 ?
Mean rate of wet station 0.1?0.8 0.08 per
mile
Occurrence of wet station is also Poisson
?P (S) P ( no wet station in 15 mile)
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Time to next occurrence in Poisson process
Time to next occurrence T is a continuous r.v.
P (X 0 in time t)
Recall for an exponential distribution
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?T follows an exponential distribution with
parameter l n
? E(T) 1/n
If n 0.1 per year, E(T) 10 years
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Comparison of two families of occurrence models
Bernoulli Sequence Poisson Process
Interval Discrete Continuous
No. of occurrence Binomial Poisson
Time to next occurrence Geometric Exponential
Time to kth occurrence Negative binomial Gamma
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Significance of correlation coefficient
r 1.0
r -1.0
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r 0
0lt r lt1.0
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Functions of Random Variable (R.V.)

• In general, Y g(X)
• Y g(X1, X2,, Xn)
• If we know distribution of X ? distribution of Y?

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Consider M 4X 10

• Observation the distribution of y depends on
• (1) Distribution of X
• (2) g(X)

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Eq. 4.6

• E4.1

See E4.1 in text for details
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Summary of Common Results
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Summary of Common Results (Contd)
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E4.8
mD 4.2 , sD 0.3 , dD 7 mL 6.5 , sL 0.8
, dL 12 mW 3.4 , sW 0.7 , dW 21
S total load D L W
Assume D, L, W are s.i.
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E4.8 (Contd)
P(failure) P(R lt S) P(R S lt 0) P(X lt 0)
X R S
R N(mR, dRmR) where mR 1.5 mS
1.5 x 14.1 21.15
dR 0.15 ? R N(21.15, 3.17)
mR design capacity 1.5 design safety factor,
SF
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• If the target is P(F) 0.001 ? mR ? and assume
  dR 0.15

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• W weight of a truck N(100, 20)
• We are interested in the total weight of 2 trucks

?
or
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E4.10

• Assume P, B, I, and M are s.i. and log-normal
  with parameters lP, lB, lI, lM and zP, zB, zI,
  zM, respectively

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Central Limit Theorem

• S will approach a normal distribution regardless
  the individual probability distribution of Xi if
  N is large enough

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Taylor Series Approximation
First order approximation
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Observe validity of linear approx depends on

• 1) Function g is almost linear, i.e. small
  curvature
• 2) sx is small, i.e. distribution of X is narrow

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Uses

• 1. Easy calculations
• 2. Compare relative contributions of
  uncertainties allocation of resource
• 3. Combine individual contributions of
  uncertainties

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Reliability Computation

• Suppose R denotes resistance or capacity
• S denotes load or demand
• Satisfactory Performance S lt R
• PS P(S lt R) and Pf 1 - PS

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Example on Case 3 S N(5, 1)
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Reliability Based Design

• Observe for Case 1 in which R and S are both
  Normal

If b ? ? PS ? ? Reliability ? ? b Reliability
index F-1(PS)
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