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MAE 4353 Mechanical Design IIDr James A. Kidd

Module 3, Part 1: 8/25/14

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Statistics & Fatigue Statistics Review

Fatigue Fatigue in Metals Fatigue Failures Fatigue Life Methods ( Stress-Life, Strain-Life, Linear-Elastic Fracture Mechanics) Endurance Limit Fatigue Strength Modifying Factors Stress Concentration and Notch Sensitivity Fluctuating Stress Characterization Fatigue Failure Criteria Loading Mode Combinations Fluctuating Stresses and Cumulative Fatigue Damage Problems

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Probability Distribution - Normal

Normal (Gaussian) Probability Density Function (PDF)

Shorthand:

Probability Distribution: Measure of uncertainty of a given quantity or measurement

Fatigue life and component wear data are noisy and must be treated accordingly forgood design and analysis

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Probability Distribution Normal, cont

Transforming into standardvariable z permits use CPD tables:

No closed form for Cumulative ProbabilityDistribution (CPD) for Gaussian:

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Empirical Distribution Rule

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Weibull DistributionVariable shape; Highly useful for reliability estimates based on experimental or field data

3-Parameter Weibull Distribution:

x 0 is the minimum guaranteed valued of x

Characteristic Value of x below which lie 63.2%

of observations

Shape parameter b >0Controls skew(large to right, small to left)

Reliability is complement to CPD:

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Weibull Distribution, cont

Solving for x:

x - Value at which specified reliability could be expected(given distribution defined by x0 , and b )

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Error Propagation

=Coefficient of Variation:

Table 20-6

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Error Propagation ExampleGiven force and area measurements of a rod under tension with normalprobability distributions and means of F = 20,000 psi and A = 0.5 in 2 andcoefficients of variation of CF = 0.05 and CA = 0.05, determine the mean andstandard deviation of the resulting direct stress. If the material has a yield stressallowable of 43,000 psi, comment on the implications of the results.

Going further: What are the implications of material property statistical variation?MMPS Handbook properties:

A Basis 99% samples exceed value to 95% confidenceB Basis 90% samples exceed value to 95% confidence

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Fatigue

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Fatigue in MetalsFailure - Under varying load

- Well below ultimate strength

- Possible with little structural deformation, i.e., suddenly

Cracks typically begin at discontinuities, for example:Rapid cross section changesKeywaysStress concentration pointsTool marks, scratches, etc.

Surface pits from contact or corrosionMaterial flaws

General fatigue classifications:Low Cycle: N10 3 cyclesHigh Cycle: N>10 3 cycles

Progression:Stage I: Micro-crack initiation and initial growthStage II: Micro-crack to macro-crack, beach mark bandsStage III: Rapid growth to failure

http:// youtu.be/DykiHVrVkKgTWI Fatigue Testing

http:// youtu.be/GPYw8hRkyVAFatigue Crack Propagation

http://youtu.be/DykiHVrVkKghttp://youtu.be/DykiHVrVkKghttp://youtu.be/GPYw8hRkyVAhttp://youtu.be/GPYw8hRkyVAhttp://youtu.be/GPYw8hRkyVAhttp://youtu.be/DykiHVrVkKg

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Fatigue-Life Methods - Introduction Stress-Life Method

Most traditional and represents high-cycle fatigue adequately Least accurate for low-cycle fatigue Strain-Life Method

More detailed analysis of localized plastic deformations Better for low-cycle fatigue estimates Multiple compounding idealizations so significant

uncertainties Discuss due to insight into nature of fatigue

Linear-Elastic Fracture Mechanics (LEFM)

Fracture mechanics crack growth as function of stressintensity Typically implemented via computer codes

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Stress-Life Method Fatigue strength from test data Widely used method:

Moores rotating beam for pure bending ofclose tolerance, highly polished specimens

Note statistical nature of the results

Testing of material andcomponents may berequired

Example with endurancelimit, S e

If no clear endurance limit,

S e must include number ofcycles (say 10 8)

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Endurance Limit Strength

Simplified relationship

for endurance limitstrength in steels:

For test specimen NOT actual component!

Fig 6-17

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Endurance Limit Modifying Factors

Marin Equation:

Endurance strength in real life can (will!) differ from lab test data using carefullyprepared specimens in controlled environments

Real effects include: Material, manufacturing, environment, design,

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Factors, cont.Surface Factor k a : Test specimens are highly machined, highly polished and in a protected environment Real components may have a variety of finishes (ground, machined, etc.)

Finish a, kpsi bGround 1.34 -0.085

Machined or cold-drawn 2.70 -0.265

Hot-Rolled 14.4 -0.718

As-Forged 39.9 -0.995

Sut , kpsi150

k a

0.875

0.7160.394

0.273

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Factors, cont.Size Factor k b : Real components vary in size from the test specimens

For bending and torsion:

For axial loading (no effect):

Equivalent diameters:For non-rotating cylinders:

For rectangular cross-sections:

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Factors, cont.Loading Factor k c : Accounts for differences in component loading vs. test loading When torsion is combined with bending or axial, use k c=1

Temperature Factor k d : For reduced temperatures, brittle fracture is a higher risk For elevated temperatures a model for a steel is:

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Factors, cont.Reliability Factor k e : Fatigue strength values usually expressed as means

Data indicates an 8% typical standard deviationReliabilty, % z a k e

50 0 1

90 1.288 0.897

95 1.645 0.868

99 2.326 0.814

99.9 3.091 0.753

99.99 3.719 0.702

99.999 4.265 0.659

99.9999 4.753 0.620Miscellaneous Effects Factor k f : For anything else, like

Corrosion, coatings,

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Endurance Strength ExampleWhat is the endurance strength of the machined shaft given below if a 99.99%reliability is desired?

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Assignments Friday (8/22) by 2200:

Complete and submit Resume Slide Assignment to D2L Dropbox By class time Monday (8/25):

Read Shigley Chapters 20 (stats) and 6 (Fatigue) Complete online Stats & Fatigue Quiz Download Stats/Fatigue Problem set (conditional on quiz) Upload Problem Set #1 (Elasticity)

Wednesday (8/27): Fatigue, continued + guest Speaker

Friday (8/29): Jet Engine Cutaway Assessment at DML 3 groups: 0900, 0945, 1030 sign up Wednesday

Monday (9/1): Labor Day No Class! Wednesday (9/3): Finish Fatigue, Introduce Project Phase I By class time Friday (9/5):

Review brainstorming videos on D2L project Phase I folder

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