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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