Lecture 2: Introduction to Uncertainty

CHOI Hae-Jin

School of Mechanical Engineering

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Contents

• Sources of Uncertainty

• Deterministic vs Random

• Basic Statistics

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Uncertainty

• Uncertainty is the information/knowledge gap between what is known and what needs to be known for optimal decision, with minimal risk. (Katzan, 1992)

• While completely eliminating uncertainty in engineering design is not possible, reducing and mitigating its effects have been the objectives of the emerging field of uncertainty management.

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Design Problems under Uncertainty

• Risk: likelihood, expressed either as a probability or as a frequency, of a hazard

• Reliability: a measure of the capacity of a part or a system to operate without failure in the service environment: e.g., reliability of 0.999 implies that there is probability of failure of 1 part in every 1000

• Safety: relative protection from exposure to hazards. A thing is safe if its risks are judged to be acceptable.

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Sources of Uncertainty

• Effect of variation on material properties

• Effect of manufacturing tolerance

• Effect of nearby assembly such as welding

• Intensity and distribution of loading

• Validity of mathematical model used to represent reality

• Influence of time on strength and geometry

• Effect of corrosion and wear

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Example of Uncertainty

Deflection by the load (P) is

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E: Modulus of elasticityI : Moment of inertiaL: length of beam

P

3

3

PL

EI (16)

Sources of Uncertainty

• Effect of variation on material properties

• Effect of manufacturing tolerance

• Intensity and distribution of loading

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3

3

L

EI

P

3

3

PL

I

E

3

3

P

EI

L

*random variable by the source of uncertainty is in bold face

Variability in Material Properties

Cx=σx/µx

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Variable (x) Typical Cx

Modulus of elasticity of metals 0.05

Tensile strength of metals 0.05

Yield strength of metals 0.07

Buckling strength of columns 0.15

Fracture toughness of metals 0.15

Cycles to failure in fatigue 0.50

Design load in mechanical components 0.05-0.15

Design load in structural systems 0.15-0.25

Table 5.4 Typical Value of Coefficient of Variation

Variability in Material Properties

• Fracture and fatigue properties show greater than the static tensile properties of yield strength and tensile strength materials exhibit variability

• Although certainly not all mechanical properties are normally distributed, a normal distribution is a good first approximation that usually results in a conservative design.

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Deterministic vs Random Variable

• Deterministic variable: a parameter that can be characterized as a discrete value.

• Random variable: a parameter but that cannot be characterized as a discrete value, but a distribution and statistical parameters; it is denoted as bold face, such as x

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Basic Probability Using the Normal Distribution

• Many physical measurements follow the symmetrical, bell-shaped curve of the normal, or Gaussian distribution.

• The equation of curve is

• where f(x) is the height of the frequency curve corresponding to an assigned value x, µ , is the mean of the population, and σ is the standard deviation of the population

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21 1

( ) exp22

xf x

(5.1)

Standard Normal Distribution

• In order to place all normal distributions on a common basis in a standardized way, the normal curve is often expressed in terms of the standard normal distribution.

• Standard normal distribution is a normal distribution with µ= 0 and σ=1. Its probability density function has the notation of , and is given by

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21( ) exp

22

xx

( )x

(5.2)

Standard normal distribution

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x

µ=0

σ=1 σ=1

( )x

21( ) exp

22

xx

N(µ, σ2) ~ N(0, 1)

Figure 5.1

Cumulative normal distribution

• Cumulative distribution function, is area under the standard normal distribution from - to x.

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xx 0

0.5

1

( ) ( ) ( )x

x y dy P y x

y

( )y

( )x

( )x 1 ( )x

( )x

0

Figure 5.2

Table 5.1 Cumulative Distribution Function of Standard Normal Distribution

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Table 5.1 Continued…

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Cumulative Standard Normal Distribution

• The symmetry of the standard normal distribution about zero implies that if the random variable Z has a standard normal distribution, then

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1 ( ) ( ) ( ) ( )x P Z x P Z x x

x-x

1 ( )x( )x

Figure 5.3

Cumulative Standard Normal Distribution

• x is the critical point of (1-α)x100 percentile of the distribution

• For example, since the 95-percentile of the standard normal distribution is x=1.645, the 95-percentile of a N(3,4) distribution is

• µ+1.645σ= 3 + (1.645 x 2) = 6.29

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α x

0.10 1.282

0.05 1.645

0.025 1.960

0.01 2.326

0.005 2.576x

1-Ф(x)=α

Table 5.2

Probability Calculation of General Normal Distribution

• Very important general result is that if

then transformed random variable

has a standard normal distribution• This result indicates that any normal distribution

can be related to the standard normal distribution by appropriate scaling and location change

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2~ ( , )X N

XZ

(5.3)

Probability Calculation of General Normal Distribution

• Probability relationship between a normal distribution and the standard normal distribution is

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2~ ( , )X N ~ (0,1)Z N

a b b

a

( )a b b a

P a X b P Z

(5.4)

Probability Calculation of General Normal Distribution

• In general, if , notice that

• From the cumulative standard distribution table, when c=1 this probability is about 68%, when c=2 this probability is about 95%, and when c=3 this probability is about 99.7%.

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2~ ( , )X N

( ) ( )

, ~ (0,1)

P c X c P c Z c

where Z N

(5.5)