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Dealing with Uncertainty in Statics

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Outline

• Uncertainty in statics• Why consider uncertainty• Basics of uncertainty• Uncertainty analysis in statics • Examples• Conclusions

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Uncertainty in Statics: An Example • Given: counterclockwise, , and • Find: • Solution

• Everything is modeled perfectly.

• In reality , , and the lengths are all random.• So is - It fluctuates around 663.0 N. • How do we know the fluctuation?

3 0 °

𝐹 2

6 0°

A

1.5 m 3.5 m

4 m

𝑀 𝐴

𝐹 1

4 5 °

𝐹 3

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Where Does Uncertainty Come From?

• Manufacturing Imprecision

- Dimensions of a structure

- Material properties• Environment

- Loading

- Temperature

- Different user conditions

- Wind, snow, etc.

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Why Consider Uncertainty?

• We know the true solution• We know the effect of uncertainty• We can make more reliability decisions• We can predict the chance of failure

If we consider uncertainties in the above example, we can design better cost-effective beams with sufficient reliability against failures with respect to strength, stiffness, etc.

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How Do We Model Uncertainty?

• Measure X = ten times, we get samples

X=(345.1,340.4,336.6,342.8,356.3,363.5, 351.9,349.3,342.4,355.7) N• How do we use the samples?• Average

(We can use excel)

3 0 °

𝐹 2

6 0°

A

1.5 m 3.5 m

4 m

𝑀 𝐴

𝐹 1

4 5 °

𝐹 3

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How Do We Measure the Dispersion (Uncertainty)?

For the deviation from , we could use and , • But . • To avoid 0, we use ; • To have the same unit as , we use • To make the estimate unbiased, we actually use

standard deviation: .• Now we have for (We can use Excel)

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More About Standard Deviation (std)

• It indicates how data spread around the average

• It is always non-negative• High std means

– High dispersion– High uncertainty– High risk

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Probability Distribution• With more samples, we can draw a

histogram.

• If the vertical axis is the frequency and the number of samples is infinity, we get a probability density function (PDF) .

• The probability of .

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

10

20

30

40

50

60

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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Normal Distribution

• – mean value, can be estimated by

• How to estimate has been discussed

• is called cumulative distribution function (CDF)

1.4 1.6 1.8 2 2.2 2.4 2.6 2.80

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

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How to Calculate

• Use Excel–NORMDIST(x,mean,std,cumulative)–Cumulative=true

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Example 1

• The moment (in N∙m) acting on a beam was recorded 20 times.

• Assume the moment is normally

distributed.• Questions

- Mean =? Std = ?

- What is the probability that the

beam has a moment greater than

1192.3 N∙m?

1 1198.52 1192.53 1188.14 1190.25 1195.16 1198.77 1196.38 1190.79 1193.910 1192.211 1191.512 1199.213 1197.714 1194.115 1196.916 1193.017 1197.518 1194.919 1199.120 1192.6

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Solution: Use Excel• average moment

by AVERAGE

by STDEV• For 62

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More Equations

• independent– The occurrence of does not affect that of

• are constants• Then

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Example 2A company plans to design a picnic table, which has a weight of = 75 lb and a center of gravity at . If a man weighing = 215 lb has a center of gravity at and sits down in the position shown, determine the vertical reaction at each of the two legs B. Will the table tip over? Neglect the thickness of the legs.

GM

GT

7 in 22 in 22 in

A B

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Example cont.Solution

Since is positive, the table will not tip over.

7 in 22 in 22 in

A B

𝑊 𝑇

𝑊𝑀

2𝑁 𝐴 2𝑁𝐵

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If Consider Uncertainty

• Due to uncertainty in users lb

Due to manufacturing uncertainty lb

and are independent.• What is the probability that the table will tip

over?

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Solution:

where lb and lb

Calculate and

If the table tips over, . The probability is therefore

The probability is about 2.4%.

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Conclusions• Uncertainty always exists in applications of

statics. • We can use a probability distribution, including

its mean and standard deviation , to model uncertainty or randomness.

• Uncertainty may have a high impact on performance.

• If uncertainty is large, we should consider it during analysis and design in order to make better decisions.