1. 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 Materials
Student 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)

2. 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/Statistical
If 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)( ∞→= N
N
AN
AP
3-Geometric
All the possible outcomes are equally likely.
Total possibilities are infinite.
4-Axiomatic (Kolmogorov)
Probabilities follow certain well-defined
rules of mathematics.

3. • 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 Quote
The most important questions of life are indeed, for the most part, really only problems of probability
- Pierre Simon Laplace - Théorie Analytique des 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”.

4. 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 Quote
There is no problem in all mathematics that cannot be solved by direct counting - Ernest Mach 4
centuries before.

5. Author: Jennifer E. Smith
Release Date: January 2, 2012
Quirks of timing play out in this
romantic and cinematic novel
about family connections,
second chances, and first
The Statistical Probability of Love at First Sight: (literature)
Probability Quote
The theory of probability as a mathematical discipline can and should be developed from axioms
in exactly the same way as geometry and algebra – Kolmogorov, 1933
5
second chances, and first
loves. Set over a twenty-four-
hour-period, Hadley and Oliver's
story will make you believe that
true love finds you when you are
least expecting it.

6. Application: Design of Composite Materials (technical)
Probability Quote
The 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

7. Example1:Probability based design of a composite C-channel for a wind-mill
C-channel
Issues:
Deformation,
Residual stresses,
Probability Quote
Misunderstanding of probability may be the greatest of all impediments to scientific literacy
-S. J. Gould
7
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

8. 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 for a wind-mill
Probability Quote
It is remarkable that a science which began with the consideration of games of chance should have
become the most important object of human knowledge - Théorie Analytique des Probabilités, 1812
8
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

9. 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 for a wind-mill
Probability Quote
The record of a month’s roulette playing at Monte Carlo can afford us material for discussing
the foundations of knowledge- Karl Pearson
9
This predicted probabilistic model is
essential to evaluate the quality of
the C-channel prior to manufacturing.

10. 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 for 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 Quote
There are two times in a man’s life when he should not speculate; when he can’t afford it,
and when he can - Mark Twain
10
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.

11. The distribution function for two parameter Weibull distribution;
Example 2: Weibull Fracture (Tensile) Analysis of Carbon Composite 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 Quote
The most important questions of life are indeed, for the most part, really only problems of probability
- Pierre Simon Laplace - Théorie Analytique des 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)-cc
-F(x;b,c)
lnln
1
1
lnln =












12. The median rank is a good estimator of distribution function as;
(4)
40
30
c)b,;(x )(
.n
.i
F i
+
−
=
Slope = c = 17.44
Y-intercept = b= 510.76
Example 2: Weibull Fracture (Tensile) Analysis of Carbon Composite Panel
Probability Quote
Probability 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

13. 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 = 0.90
UTS ≥ 448.92
Example 2: Weibull Fracture (Tensile) Analysis of Carbon Composite Panel
Probability Quote
There 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

14. THANKS
谢谢谢谢谢谢谢谢
Probability Quote
Dilbert 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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