Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

94
Why need probabilistic Why need probabilistic approach? approach? Rain probability Rain probability How does that affect our How does that affect our behaviour? behaviour? ? ?
  • date post

    20-Dec-2015
  • Category

    Documents

  • view

    214
  • download

    0

Transcript of Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Page 1: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Why need probabilistic Why need probabilistic approach?approach?

Rain probabilityRain probability

How does that affect our behaviour?How does that affect our behaviour?

? ?

Page 2: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Uncertainties in Uncertainties in EngineeringEngineering

Natural HazardsNatural Hazards

Material PropertiesMaterial Properties

Design ModelsDesign Models

Construction ErrorsConstruction Errors

Page 3: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Absolute Safety Not Absolute Safety Not GuaranteedGuaranteed

Engineers need to:Engineers need to:

model, analyze, update model, analyze, update

uncertaintiesuncertainties

evaluate probability of failureevaluate probability of failure

Page 4: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

QuestionsQuestions

What is acceptable failure What is acceptable failure probability?probability?

-- stadium vs shedstadium vs shed

Page 5: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

QuestionsQuestions

Should one want to be Should one want to be conservative if a perfectly safe conservative if a perfectly safe system is possible?system is possible?

-- overbooking in airlinesoverbooking in airlines

-- parking permitsparking permits

Page 6: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

QuestionsQuestions

Should one minimize risk if Should one minimize risk if money is not a problem?money is not a problem?

-- system consideration – system consideration –

e.g.dame.g.dam

Page 7: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Trade-off Decision AnalysisTrade-off Decision Analysis

Risk vs. consequenceRisk vs. consequence

System riskSystem risk

Page 8: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Formal analysis of Formal analysis of uncertainties and probabilityuncertainties and probability

• Not all problems can be solved by analysis Not all problems can be solved by analysis of dataof data

• Set TheorySet Theory

• Sample spaceSample space: collection of all possibilities: collection of all possibilities

• Sample pointSample point: each possibility: each possibility

• EventEvent: subset of sample space: subset of sample space

• Probability TheoryProbability Theory

Page 9: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Union: Union: eithereither E E1 1 or Eor E2 2 occuroccur

EE11∪∪EE22

Intersection:Intersection:bothboth E E1 1 and Eand E2 2 occuroccur

EE11∩∩ E E22 or E or E11 E E22

Page 10: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

ExamplesExamples

A B

No communication between A and B = E1E2

CB

A

No communication between A and B = E3∪∪E1E2

1

2

EE11 = road 1 closed = road 1 closedEE22 = road 2 closed = road 2 closed

EE33 = road 3 closed = road 3 closed1

23

Page 11: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Example - pair of footingsExample - pair of footings

1 2

EE11 = 1 settles = 1 settles

ĒĒ11 = 1 does not = 1 does not

settlesettle

EE22 = 2 settles = 2 settles

ĒĒ22 = 2 does not = 2 does not

settlesettle

Settlement occurs = E1∪∪E2

Tilting occurs = E1ĒĒ2∪∪ĒĒ1E2

Page 12: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

de Morgan’s rulede Morgan’s rule

EE1 1 = pipe 1 breaks= pipe 1 breaks

EE22 = pipe 2 breaks = pipe 2 breaks

1 2

Water Supply

E = failure in water supply = EE = failure in water supply = E11∪∪EE22

no failure in water supply = no failure in water supply = ĒĒ = =

EE11∪E∪E22

Page 13: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

2121EEEE

nEEEEEE n ....... 2121

Event of “no failure”

Extension to n events

Page 14: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

de Morgan #2de Morgan #2

2121 EEEE

nn EEEEEE ........ 2121

Page 15: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Basis of Probability Basis of Probability EstimationEstimation

a)a) Subjective assumption e.g. P(Q) = 1/2Subjective assumption e.g. P(Q) = 1/2

b)b) Relative frequency e.g. Relative frequency e.g.

P(Q)=502/1000P(Q)=502/1000

c)c) Bayesian (a)+(b) Bayesian (a)+(b) judgment + limited observation judgment + limited observation

Page 16: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Probability of UnionProbability of Union

in generalin general

E1 E2

1 2( )P E E

1 2 1 2( ) ( ) ( )P E P E P E E

Page 17: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Using de Morgan’s ruleUsing de Morgan’s rule

)(1)( 321321 EEEPEEEP

)(1 321 EEEP

P (intersection) conditional

probability

Page 18: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

( ) ( | ) ( )P AB P A B P B ( ) ( | ) ( )P AB P B A P A

( ) ( | ) ( )P ABC P A BC P BC

( | ) ( )P B C P C

or or

Page 19: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Statistical independence

if E1 and E2 are s.i.

2 1 2( ) ( )P E E P E

1 2 1( ) ( )P E E P Eor

s.i.

Page 20: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Example:

E2 = flood in 廣西 on June E3 = flood in 哈爾濱 on June

E1 = flood in 廣東 on June

P(E1) = 0.1; P(E2)=0.1; P(E3) = 0.1

121 3.0| EPEEP E1 and E2 are not s.i.

1.0| 31 EEPE1 and E3 are s.i.

1EP

Page 21: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

1 2 1 2 2( ) ( ) ( )P E E P E E P E

1( )P Eif E1 and E2 are s.i.

1 2 3 1 2 3( ) ( ) ( ) ( )P E E E P E P E P Eif all are s.i.

Page 22: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

s.i. and m.e. (mutually exclusive)

if E1 and E2 are m.e.

1 2( ) 0P E E

1 2 1( ) ( )P E E P E

if E1 and E2 are s.i.

Page 23: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

B

C

A

2

1

3

P(E1)=2/5

P(E2)=3/4

P(E3)=2/3

P(E3|E2)=4/5

P(E1|E2E3)=1/2

a) P(go from A to B through C)

32EEP 2 3( ) ( )P E P E

5

3

4

3

5

4

3 2 2( ) ( )P E E P E

E1 : ① is openP 2.15

Page 24: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

b)

P(go from A to B)

132 EEEP

132132 EEEPEPEEP

1 2 3 2 3

3 2( | ) ( )

5 5P E E E P E E

7.0 1/2 3/5

Page 25: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

T.O.T (Theorem of Total Probabilities)

Bayes theorem

AP

EPEAPAEP jj

j

||

P(A) = P(A|E1)P(E1)+P(A|E2)P(E2)+…+P(A|En)P(En)

Ei’s are m.e. and c.e.

Page 26: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

7.0GP 3.0GP

good enough for construction

2.0|;8.0| GTPGTP

9.0|;1.0| GTPGTPpositive

E 2.30 aggregate for construction

engineer's judgment based on geology and experience

crude test

reliability (or quality) is as follows:

not a perfect test

Page 27: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

After 1 successful test, what is P(G)?

TP

GPGTPTGP

||

3.01.07.08.0

7.08.0

95.0

( | ) ( )

( | ) ( ) ( | ) ( )

P T G P G

P T G P G P T G P G

Page 28: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

After another successful independent test, P(G)?

GPGTPGPGTP

GPGTPTGP

||

||

22

22

05.01.095.08.0

95.08.0

993.0

Page 29: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

What if the two tests were performed at the same time?

)()|()()|(

)()|()|(

2121

2121

GPGTTPGPGTTP

GPGTTPTTGP

1 2

1 2 1 2

( | ) ( | ) ( )

( | ) ( | ) ( ) ( | ) ( | ) ( )

P T G P T G P G

P T G P T G P G P T G P T G P G

0.7

0.30.7

3.1.1.7.8.8.

7.8.8.

993.0

P(G)

UST

HKU

0.7

0.3

After 1 test

0.95

0.77

After 2 tests

0.993

0.965

… 5

1.0000

0.9999

Page 30: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Random variables

A device to:

a) formalize description of event

b) facilitate computation of

probability

Page 31: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

PMF PX(x) FX(x)

PDF fX(x) FX(x)

CDF

( )( ) X

X

dF xf x

dx

Page 32: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Main descriptors of R.V.

The PMF or PDF completely define the r.v.

Descriptors give partial information about the r.v.

Page 33: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Mean value

Define = E(X)

= expected value of X or mean value of X

( )i X ii

x P xa measure of central tendency a measure of central tendency

Page 34: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Measure of spread

Standard deviation X

X dimensionless %

range

( )Var X

X

X

Page 35: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Expected value of function

( ) ( )XE X xf x dx

recall

( ) ( ) ( )XE g X g X f x dx

Page 36: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Recall2( ) [ ( )] ( )XVar X x E X f x dx

2 2( ) ( ) ( )Var X E X E X

After some algebra,

2 ( )Xx f x dx

Page 37: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

0.00

0.05

0.10

0.15

0.20

0.25

-5 0 5 10 15 20

fX(x)

x

Normal distribution

= 5

X : N (, )

N (5, 2)

x

Page 38: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Effect of varying parameters ( & )

fX(x)

x

A

B

C

for C

for B

Page 39: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

S: N (0,1)

Standard normal distribution

0.00

0.10

0.20

0.30

0.40

0.50

-6 -4 -2 0 2 4 6

fX(x)

x

Page 40: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

(0) 0.5 (1) 0.8413

(2) 0.97725 ( 1) 1 (1) 0.1587

(4) 1 0.00003167 0.9999683

-5 -4 -3 -2 -1 0 1 2 3 4 5

21

21( )

2

a sa e ds

a

Page 41: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Page 380 Table of Standard Normal Probability

Page 42: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Example: retaining wall

x

F

Suppose X = N(200,30)

(230 260)

260 200 230 200

30 30

(2) (1)

0.977 0.841 0.136

P x

Page 43: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

If the retaining wall is designed such that the reliability against sliding is 99%,

How much friction should be provided?

1

( ) 99%

200 2000.99

30 30

200(0.99)

30

P x F

F

F

1200 30 (0.99) 270F

2.33

Page 44: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Lognormal distribution

Parameter

0 2 4 6 8

fX(x)

x

Page 45: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

21ln

2

2

2 22

ln 1 ln 1

Parameters

for 0.3,

ln mx

Page 46: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Probability for Log-normal distribution

ln( )

aP X a

If a is xm, then is not needed.

ln ln( )

b aP a X b

Page 47: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Other distributions

Exponential distribution Triangular distribution Uniform distribution Rayleigh distribution

p.224-225: table of common distribution

Page 48: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Exponential distribution

x

fX(x)

( ) xXf x e x 0

1( )E X

21

( )Var X

100%X

Page 49: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Beta distribution1 1

1

( ) ( ) ( )( )

( ) ( ) ( )

q r

X q r

q r x a b xf x

q r b a

a x b

0.0

0.1

0.2

0.3

0 2 4 6 8 10 12x

fX(x)q = 2.0 ; r = 6.0

a = 2.0 b = 12

probability

Page 50: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1

Standard beta PDF

q = 1.0 ; r = 4.0

q = r = 3.0 q = 4.0 ; r = 2.0

q = r = 1.0

x

fX(x)

(a = 0, b = 1)

Page 51: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Bernoulli sequence

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

trials

SF

p = probability of a success

Page 52: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

SF

x = number of successp = probability of a success

P ( x success in n trials)

= P ( X = x | n, p) (1 )

( 0,1,2,..., )

x n xnp p

x

x n

Binomial distribution

Page 53: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Examples

Number of flooded years

Number of failed specimens

Number of polluted days

Page 54: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Example:

Given: probability of flood each year = 0.1

Over a 5 year period

1 45( 1) 0.1 (0.9) 0.328

1P X

P ( at most 1 flood year) = P (X =0) + P(X=1)

= 0.95 + 0.328

= 0.919

Page 55: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

P (flooding during 5 years)

= P (X 1)

= 1 – P( X = 0)

= 1- 0.95

= 0.41

Page 56: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

For Bernoulli sequence Model

No. of success binomial distribution

Time to first success geometric

distribution

E(T) =1/p = return period

Page 57: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Significance of return period in design

Suppose a bldg is expected to last 100 years and if it is designed against 100 year-wind of 68.6 m/s

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

Service life

design return period

Page 58: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

P (1st exceedence of 68.6 m/s in 100th year)

= 0.99990.01 = 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

Page 59: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

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)

= 0.995990.005 = 0.003

Page 60: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

P (no exceedence of 70.6 m/s within return period of design)

= 0.995200 = 0.367

P (no exceedence of 70.6 m/s within a service life of 100 years)

= 0.995100 = 0.606 > 0.366

Page 61: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

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

Page 62: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

N (72,8)Example

V100

0.01

Design for return period of 100 years:

p = 1/100 = 0.01

100( ) 0.99P V V

100 720.99

8

V

V100 = 90.6 mph

Annual max wind velocity

Frequency

Page 63: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Suppose we design it for 100 mph, what is the corresponding return period?

( 100)

100 721

8

1 (3.5)

0.000233

P V

T 4300 years

Alternative design criteria 1

Page 64: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Pf = P (exceedence within 100 years)

= 1- P (no exceedence within 100 years)

=1- (1-0.000233)100 = 0.023

Probability of failure

Page 65: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

1x n xn t t

x n n

P ( x occurrences in n trials)

= limn

( )

!

xtt

ex

x = 0, 1, 2, …

Poisson distributionPoisson distribution

Page 66: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

P 3.42

Service stations along highway are located according to a Poisson process

Average of 1 station in 10 miles = 0.1 /mile

P(no gasoline available in a service station)

( ) 0.2P G

Page 67: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

(a) P( X 1 in 15 miles ) = ?

0 1

0 1.5 1 1.5

( 0) ( 1)

( ) ( )

0! 1!

(1.5) (1.5)

0! 1!0.223 0.335

0.558

t t

P X P X

t e t e

e e

No. of service stations

Page 68: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

(b) P( none of the next 3 stations have gasoline)

0 3

( 0 | 3, )

( 0 | 3,0.8)

3(0.8) (0.2)

0

0.008

P Y p

P Y

binomial

No. of stations with gasoline

Page 69: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

(c) A driver noticed the fuel gauge reads empty; he can go another 15 miles from experience.

P (stranded on highway without gasoline) = ?

P (S)( | 0) ( 0) ( | 1) ( 1)

( | 2) ( 2) ......

P S X P X P S X P X

P S X P X

No. of station in 15 miles

binomial Poisson

Page 70: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

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

Page 71: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Alternative approach

Mean rate of service station = 0.1 per mile

Probability of gas at a station = 0.8

Mean rate of “wet” station = 0.10.8 = 0.08 per mile

Occurrence of “wet” station is also Poisson

P (S) = P ( no wet station in 15 mile)0

0.08 15 1.2(0.08 15)0.301

0!e e

Page 72: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Time to next occurrence in Poisson process

Time to next occurrence = T is a continuous r.v.

( ) ( ) ( ) 1 ( )T T

d d df t F t P T t P T t

dt dt dt

( )P T t = P (X = 0 in time t)te

( ) 1 t tT

df t e e

dt

Recall for an exponential distribution

( ) xXf x e

Page 73: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

T follows an exponential distribution with parameter =

E(T) =1/

If = 0.1 per year, E(T) = 10 years

Page 74: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

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

Comparison of two families of occurrence models

Page 75: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Significance of correlation coefficient

= +1.0 = -1.0

y: strength

x: Length

GlassGlass

y: elongation

x: Length

SteelSteel

Page 76: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

= 0 0< <1.0

y: ID No

x: height

y: weight

x: height

Page 77: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Functions of Random Variable Functions of Random Variable (R.V.)(R.V.)

In general, Y = g(X)

Y = g(X1, X2,…, Xn)

If we know distribution of X distribution of Y?

M = 4X + 10

X5

22

M1)

X

b

b = Xtan 2)

cost of delay = aX23) where X – length of delay

Page 78: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Consider M = 4X + 10Consider M = 4X + 10

Observation: the distribution of y depends on

(1) Distribution of X

(2) g(X)

X654

0.6

0.2 0.2

M343026

0.6

0.2 0.2

X654 M343026

wider distribution

Page 79: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

2

11

2

1

2

2 3 2

1 1 2 3 3exp 2

2 22 2

1e 0,1

2

, 0,1

y x x

y

dgf y f g f y

dy

y

N

xif x N y N

Y X X

X YX

Eq. 4.6Eq. 4.6

E4.1

See E4.1 in text for details

3

2

xy

,

3, 2x N

1 2 3 2dx

g x ydy

X

21

21

2

x

xf x e

X

where

1

1y x

dgf y f g

dy

Y X

For monotonic For monotonic function g(X)function g(X)

Page 80: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

if X1 and X2 are Poisson with mean rates 1 and 2 respectively Z is Poisson with z = 1 + 2

(see E4.5 on p. 175)

1 2Z X X

if X is N(,) Y is N(a + b, a)Y aX b ( 0)a

if X is N(,) Y is N(a, a)

if X is LN(,) Y is LN(ln a + , )Y aX

( 0)a

Summary of Common ResultsSummary of Common Results

Page 81: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Summary of Common Results Summary of Common Results (Cont’d)(Cont’d)

if X1 and X2 are N(,) and N(,) respectively Z is N(z,z)where

if X1 and X2 are s.i. 12 = 0

1 1 2 2Z a X a X

2 2 2 21 1 2 2 1 2 12 1 22Z a a a a

if Xi = N(i,i) ; i = 1 to n Z is N(z,z)where1 1 2 2

... n n

Z a X a X

a X

1 1 2 2 ...Z n na a a

2 2Z i i

i

a correlation terms

1 1 2 2Z a a

Page 82: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

E4.8E4.8

S = D + L + W

Column with capacity, R

D = 4.2 , D = 0.3 , D = 7%L = 6.5 , L = 0.8 , L = 12%W = 3.4 , W = 0.7 , W = 21%

S = total load = D + L + W

S = D + L + W = 14.1S = 0.32 + 0.82 + 0.72 =1.1

a) P(S > 18) = 1 –

= 1 – (3.55) = 0.000193

18 14.1

1.1

Assume D, L, W Assume D, L, W are s.i.are s.i.

Page 83: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

E4.8 (Cont’d)E4.8 (Cont’d)P(failure) = P(R < S)

= P(R – S < 0)= P(X < 0)

R = N(R, RR)where R = 1.5S

= 1.5 x 14.1 = 21.15R = 0.15 R = N(21.15, 3.17)

X = -14.1 + 21.15 = 7.05X = 1.12 + 3.172 = 3.36

P(F) = = (-2.1) = 0.0180 7.05

3.36

X = R – SX = R – S

RR – design capacity – design capacity

1.5 – design safety factor, SF1.5 – design safety factor, SF

Page 84: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

If the target is P(F) = 0.001 R = ? and assume R = 0.15

Recall:X = R - 14.1

X = (0.15R)2 + 1.12

= -1(0.001) = – 3.09

0.0225R2+1.21 = 20.8 – 2.95R + 0.105R

2

0.0825R2 – 2.95R +19.59 = 0

R = 8.812 or 26.9

2 2

14.1

0.15 1.1R

R

2

22 2 14.1(0.15 ) 1.1 4.56 0.324

3.09R

R R

Set: P(F) = = 0.001 2 2

14.1

0.15 1.1R

R

Since R should be larger than 21.5R = 26.9

Check R = 8.812

P(F) = = (3.09) 1.0

8.812 14.1

1.7197

Page 85: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

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

trucks

or

1.

Total weight = T = 2W

Normal with

T = 2W = 200

T = 2W = 40

Total weight = T = W1 + W2

Normal with T = 200

T =

=

=

if s.i. and 1 = 2 = 20

2 21 220 20 2

2 220 2020 2

2.

?

T

Page 86: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

E4.10E4.10

Sand Footing

P

S

Sand Property – M Footing Property – B and I

PBIS

M

Assume P, B, I, and M are s.i. and log-normal with parameters P, B, I, M and P, B, I, M, respectively

c.o.v.

P B I

M

1.0 6.0 0.6

32.0

0.10 0

0.10 0.15

Page 87: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Central Limit TheoremCentral Limit Theorem

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

0.5

0.7

0.4

0.2

0.6

0.3

0.1

-1 1

P N = 1 S = X1

0.5

0.7

0.4

0.2

0.6

0.3

0.1

-2 0 2

P N = 2 S = X1 + X2

0.5

0.7

0.4

0.2

0.6

0.3

0.1

2 -4 -2 0 4 8 6 10

P N = 10 S = X1 + X2 +…+X10

P

0.5

0.7

0.4

0.2

0.6

0.3

0.1

18 16 14 12 10 8 6 4 2 0 -2 -4

N = 20 S = X1 + X2 +…+ X20

Page 88: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

First order approximation:

2 2

X X X

X

X

X X

dgE g X g E X g

dX

dg dgVar g X Var X Var X

dX dX

g(X)

g’(X) (X – X)

X

g(X)

X X

Taylor Series ApproximationTaylor Series Approximation

X X X

2X

g(X) g( ) g ( ) X-

"...

2 X

gX

x - x

X X

dgg(X) g( ) X-

dXX

Var(X)

is known and are known valuesX

X

dgg X g

dX

2 2

X X X

X

X

X X

dgE g X g E X g

dX

dg dgVar g X Var X Var X

dX dX

Page 89: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Observe validity of linear approx Observe validity of linear approx depends on:depends on:

1) Function g is almost linear, i.e. small curvature

2) x is small, i.e. distribution of X is narrow

Page 90: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

UsesUses

1. Easy calculations

2. Compare relative contributions of uncertainties – allocation of resource

3. Combine individual contributions of uncertainties

Page 91: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Reliability ComputationReliability ComputationSuppose R denotes resistance or capacity

S denotes load or demandSatisfactory Performance = {S < R}

PS = P(S < R) and Pf = 1 - PS

Case 1: If R, S are normal

where Z = S – R and Z = S2 + R

2

0 0

0 Z

Z

P S R P S R P Z

Case 2: If R, S are lognormal

where Z = S – R and Z = S2 + R

2

ln11 1 Z

Z

SP S R P P Z

R

Page 92: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Case 3: If R is discrete, S is continuous

i ii

S i R ii

P S R P S R R r P R r

F r P r

Example on Case 3

S = N(5, 1)

5 5 6 6 7 7P S R P S P R P S P R P S P R

5 5 6 5 7 50.1 0.3 0.6

1 1 1

0 0.1 1 0.3 2 0.6

0.5 0.1 0.84 0.3 0.98 0.6

0.889

r

P(R = r)

5 6 7

0.1

0.3

0.6

Page 93: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Reliability – Based DesignReliability – Based Design

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

2 2

R SS

S R

P

If PS Reliability

= Reliability index = -1(PS)

Design Design RR = = SS + + SS22 + + RR

22

Page 94: Why need probabilistic approach? Rain probability How does that affect our behaviour? ?

Example:S = N(5, 2)R = N(R, 1)

R = ?

Require Pf = 0.001 or PS = 0.999 = -1(0.999) = 3.1

Design R = 5 + 3.122 + 12 = 11.93