a review of probability and its applications shameel farhan new applied [compatibility mode]

A Review of

Probability and Statistics-its application in Composite Materials-

Outline of Presentation

Definition (basic)

Timeline (history)

Chinese Contribution (social)

1

Shameel Farhan (夏明汉夏明汉夏明汉夏明汉)PhD student in Materials Science & Engg.Specialization in Composite MaterialsStudent Number 2013410005

Chinese Contribution (social)

Statistical probability of love at first sight (literature)

Example 1:Probability based design of C-channel (technical)

Example 2:Weibull fracture analysis of composites (technical)

Population Sample

Probability

Statistics

Probability: An engine that drives statistics

Probability Quote“Suam habet fortuna rationem.” (Chance has its reasons.) Petronius (First century, A.D.) 2

2-Frequency/StatisticalIf number of trials is N and number of the occurrence of A is N(A) then:

1-Classical •All possible outcomes are equally likely.•Total possibilities are finite.

)( )(

lim)( ∞→= NN

ANAP

3-GeometricAll the possible outcomes are equally likely.Total possibilities are infinite.

4-Axiomatic (Kolmogorov)Probabilities follow certain well-defined rules of mathematics.

• 2000(BC)- Games of chance

• 1494- F. L. Paccioli wrote the first printed work addressing probability

called Summa de arithmetica, geometria, proportioni e proportionalita.

• 1550 – G. Cardano’s book “games of chance”

• 1654 - Chevalier de Méré asks Pascal gambling question and 7 letters are exchanged b/w Pascal and Fermat in a mere 4 month span

• 1657 – Christanus Huygens wrote the treatise De Ratiociniis in Aleae Ludo

• 1662 - John Graunt writes Observations on the Bills of Mortality

Timeline: (history)

Probability QuoteThe most important questions of life are indeed, for the most part, really only problems of probability

- Pierre Simon Laplace - ThéorieAnalytiquedes Probabilités, 1812. 3

• 1662 - John Graunt writes Observations on the Bills of Mortality

• 1713 – Jakob Bernoulli wrote Ars Conjecture

• 1718 - Abraham DeMoivre’s Doctrine of Chances: or, a Method of Calculating

the Probability of Events in Play is published

• 1812 – New approach first book: P. Laplace, The Analytical Theory of Probability

• 1933- Modern theory of probability (20th): Kolmogorov : Axiomatic approach

• 1950- First modern book: A. Kolmogorov, “Foundations of Probability Theory”.

Zhu Shi jie was one of the greatest Chinese mathematicians born in 13th century near Beijing (Yuan Dynasty). Two of his mathematical works have survived; “Introduction to Computational Studies” and “Jade Mirror of Four Unknowns”.

Yang worked on magic squares , binomial theorem , and Yang Hui's Triangle '. This triangle was the same as Pascal's Triangle , discovered centuries before.

Chinese Contribution

Probability QuoteThere is no problem in all mathematics that cannot be solved by direct counting - Ernest Mach 4

centuries before.

Author: Jennifer E. SmithRelease Date: January 2, 2012

Quirks of timing play out in thisromantic and cinematic novelabout family connections,second chances , and first

The Statistical Probability of Love at First Sight: (literature)

Probability QuoteThe theory of probability as a mathematical discipline can and should be developed from axioms

in exactly the same way as geometry and algebra – Kolmogorov, 19335

second chances , and firstloves. Set over a twenty-four-hour-period, Hadley and Oliver'sstory will make you believe thattrue love finds you when you areleast expecting it.

Application: Design of Composite Materials (technic al)

Probability QuoteThe excitement that a gambler feels when making a bet is equal to the amount he might win

times the probability of winning it. -Pascal 6

InputInput PDF/ANSYSPDF/ANSYS OutputOutput

• Material properties• Geometry• Boundary Conditions

• Deformation• Stresses, strains• Fatigue, creep,...

Scatter Uncertain

Probability-Based Design of Composites

Example1: Probability based design of a composite C-channel f or a wind-mill

C-channel

Issues: Deformation, Residual stresses,

Probability QuoteMisunderstanding of probability may be the greatest of all impediments to scientific literacy

-S. J. Gould7

Residual stresses, Assembly problems

Random Variables Mean COV Distribution

RCS (Resin Cure Shrinkage) 0.1 0.1 Normal

Vf (Volume fraction of fiber) 0.57 0.1 Normal

CTE (Coefficient of thermal expansion) 28.6 0.1 Normal

T (Hold temperature) 350 0.01 Normal

Step-1 : Defining random variables with mean, COV and std deviation

Probability of G>0 ( θ ≤ θ0 ) for various tolerances

θ0 (˚) Prob(G>0) θ0 (˚) Prob(G>0)

Performance function = Tolerance-Deformation

(1)

If than

If than

Probability of failure + Reliability = 1

(2)

0<G

DCG −=

)0( obPr <= GPf

0>G )0( obPrr >= GP

1r =+ PfP

Step-3: Computing reliability analysis, using RELAN with COMPRO

Step-2: Defining Performance and Probability density function

Example1: Probability based design of a composite C-channel f or a wind-mill

Probability QuoteIt is remarkable that a science which began with the consideration of games of chance should have

become the most important object of human knowledge - ThéorieAnalytiquedes Probabilités, 18128

θ0 (˚) Prob(G>0) θ0 (˚) Prob(G>0)

1.00 0.00123 1.50 0.52244

1.20 0.03818 1.55 0.63599

1.25 0.07207 1.60 0.73785

1.30 0.12490 1.65 0.82168

1.35 0.19949 1.70 0.88656

1.40 0.29498 1.80 0.95963

1.45 0.40496 2.00 0.99727

The limit-state function Eq. (1) can be rewritten

The joint probability density function over the failure region is;

),....xx,(x n21DCG −=

∫<

=0

2121,...3,2,1 ,...,),....,(G

nnn dxdxdxxxxfG

Step-4: Curve fitting using least square method we get the distribution, which is normal with mean value of 1.50°, and std deviation of 0.17°

Example1: Probability based design of a composite C-channel f or a wind-mill

Probability QuoteThe record of a month’s roulette playing at Monte Carlo can afford us material for discussing

the foundations of knowledge- Karl Pearson9

This predicted probabilistic model is essential to evaluate the quality of the C-channel prior to manufacturing.

Step-5: Sensitivity analysis of probability function to the mean value of Vf (random variable).

Case Study

Increasing Vf from 0.5 to 0.6 will shift the probability

vs 6.05.0 VfVf

Example1: Probability based design of a composite C-channel f or a wind-mill

will shift the probability distribution to the left without varying its shape resulting in probability function as;

For a given value of θ0 , an increase in Vf will increase the probability of survival;

Probability QuoteThere are two times in a man’s life when he should not speculate; when he can’t afford it,

and when he can - Mark Twain10

5.06.0 FF >

)0( obPr >G

Similarly other random variables can be changed one by one or in pairs and probability functions

can be evaluated.

The distribution function for two parameter Weibull distribution;

Example 2: Weibull Fracture (Tensile) Analysis of Carbon Compos ite Panel

Test No 1 2 3 4 5 6 7 8

UTS (MPa) 532.7 442 473 519 502.7 477 510 552

Test No 9 10 11 12 13 14 15 -

UTS (MPa) 522 439 513.6 497.5 450.9 507.3 463.5

Weibull distribution is used to calculate the fracture times and fracture strength.

0c0,b , ]c)x

(exp[1c)b,F(x; ≥≥−−= (1)

The reliability is then defined as

(2)

The parameters b and c are estimated by method of linear regression. Taking

double log of both sides of Eq. (2) we get;

(3)

Probability QuoteThe most important questions of life are indeed, for the most part, really only problems of probability

- Pierre Simon Laplace - ThéorieAnalytiquedes Probabilités, 1812.11

0c0,b , ]c)b

x(exp[1c)b,F(x; ≥≥−−=

0c0,b , ]c)b

x(exp[c)b,R(x; ≥≥−=

(b)(x)-c c -F(x;b,c)

lnln 1

1lnln =

The median rank is a good estimator of distribution function as;

(4)40

30c)b,;(x )( .n

.iF i +

−=

Slope = c = 17.44Y-intercept = b= 510.76

Example 2: Weibull Fracture (Tensile) Analysis of Carbon Compos ite Panel

Probability QuoteProbability theory is nothing but common sense reduced to calculation- Laplace 12

C< 0 a decreasing failure rate C=0 a constant failure rate C>0 an increasing failure rate

Putting the values b, c, and x = b in Eq (4)

(4)

R(510.76; 510.76, 17.44) = 0.368 = 36.8%

That is 36.8% of the tested specimens have a fracture strength of at least 510.76 MPa

0c0,b , ]c)b

x(exp[c)b,R(x; ≥≥−=

Weibull reliability distribution:

R = 0.90UTS ≥ 448.92

Example 2: Weibull Fracture (Tensile) Analysis of Carbon Compos ite Panel

Probability QuoteThere is no problem in all mathematics that cannot be solved by direct counting - Ernest Mach 13

distribution:

Conclusions : The material will fracture with 0.90 probability for a tension of ≥ 448.92 MPa

THANKS谢谢谢谢谢谢谢谢

Probability QuoteDilbert writes a poem and presents it to Dogbert:Dogbert: I once read that given infinite time, a thousand monkeys with typewriters would eventually

write the complete works of Shakespeare.Dilbert: But what about my poem?Dogbert: Three monkeys, ten minutes.

-Scott Adams, Dilbert comic stripe, May 15, 1989 14

谢谢谢谢谢谢谢谢

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