Fatigue Analysis - UTEP

Stress-Life Approach

For Extreme Environments

Fatigue Analysis

Stress-Life ApproachPresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Spring 2021

Outline

• Fatigue Loading

• Stress-Life Approach

• Unmodified Endurance Limit

• Endurance Limit Modifiers

• Mean Stress

• Solving Problems

Fatigue Loading

Fatigue Loading

• Components, structures, and vehicles are subjected to quite diverse load histories.

• Some load histories may be simple and repetitive.

• Others may be completely random.• The randomness may contain substantial portions of more deterministic

loading.

• For example, the ground-air-ground cycle of an aircraft has substantial similarity from flight to flight.

Fatigue Loading

Ground-Air-Ground Cycle of AirCraft

Repetitive

Randomness

Fatigue Loading

Offshore Structures

Aircraft

Environmental Effects

Fatigue Loading

Typical Load Histories From Actual Ground Vehicle Components

Fatigue Loading

A Typical Load History of Short-Span Bridge

Fatigue Loading

• These load histories are typical of those found in real-life engineering situations.

• Fatigue from variable amplitude is discussed in Chapter 9.

• Constant amplitude loading is introduced in this chapter.

• Constant amplitude loading is used:• To obtain material fatigue behavior/properties for use in fatigue design,

• Some real-life load histories can occasionally be modeled as essentially constant amplitude.

Sinusoidal Fluctuating Stress

Pulsating Tension

Completely Reversed (S-N Curve)

Fig. 6–23

Fluctuating stress with high frequency ripples

Nonsinusoidalfluctuating stress

Nonsinusoidalfluctuating stress

LaboratoryReal World

Shigley’s Mechanical Design

Fatigue Loading


Fatigue Loading• Def’s:

– Stress range

– Stress amplitude

–Mean stress

– Stress ratio, R

– Amplitude ratio, A

max min

2a

−=

max min

2m

+=

max minr = = −

min

max

R

=

m

aA

=

Period

Frequency

1

f =

1f

=


Fatigue Loading

• Fully Reversed: R = -1; A = ∞

• Zero to Tension: R = 0; A = 1

• Zero to Compression: R = ∞; A = -1

• R w.r.t A

• A w.r.t. R

1

1

AR

A

−=

+

1

1

RA

R

−=

+

R = -1

R = 0


Fatigue Loading

• R= -1 and R= 0 are two common reference test conditions used for obtaining fatigue properties.

• R= -1 is called the fully reversed condition since Smin= -Smax

• R = 0, where Smin = 0, is called pulsating tension.

• One cycle is the smallest segment of the stress versus time history which is repeated periodically.• Under variable amplitude loading, the definition of one cycle is not clear and

hence reversals of stress are often considered.

• In constant amplitude loading, one cycle equals two reversals.


Stress-Life Approach: (S-N)

• The Stress-Life, S-N, method was the first approach used to understand and quantify metal fatigue.

• It was the standard fatigue design method for almost 100 years.

• The S-N approach is still widely used in design applications where the applied stress is primarily within the elastic range of the material and the resultant lives (cycles to failure) are long, such as power transmission shaft.

• The Stress-Life method does not work well in low-cycle fatigue, where the applied strain have a significant plastic component.

• In low-cycle fatigue the Strain-Life method is more appropriate.

Stress-Life Approach: Typical Curve


• Typical schematic S-N curve obtained under axial load or stress control test conditions with smooth specimens.

• Constant amplitude S-N curves of this type are plotted on semi-log or log-log coordinates.

• S-N curves obtained under torsion or bending load-control test conditions often do not have data at the shorter fatigue lives (say 10^3 or 10^4 cycles and less) due to significant plastic deformation.

• Torsion and bending stress equations τ=T r / J and σ= M y /I can only be used for nominal elastic behavior.

Stress-Life Approach: Variability

• Typical variability with less variability at shorter lives and greater variability at longer lives.

• Variability in life for a given stress level can range from less than a factor of two to more than an order of magnitude.

• Variability and statistical aspects of fatigue data are discussed in Ch. 13.

Stress-Life Approach: Endurance Limit

• Fig. (a) shows a continuous sloping curve, while Fig. (b) shows a discontinuity or “knee” in the S-N curve which indicates an endurance limit.• This endurance limit has been found in only a few materials (i.e. low and medium

strength steels) between 10^6 and 10^7 cycles in non-corrosive conditions.• Most materials do not contain the endurance limit even under controlled

environments.• Under corrosive environments all S-N data have a continuous sloping curve.

• When sufficient data are available, S-N curves are usually drawn through median points and thus represent 50 percent expected failures.

S-N Diagram for Steel in Laboratory Air

( )010 , utS

Endurance Limit


S-N Diagram for Steel in Laboratory Air

( )010 , utS

Endurance Limit

Finite Life

Infinite Life

Low Cycle Fatigue(LCF)

High Cycle Fatigue


Stress-Life Approach: Definitions

• Definitions

• Cycles to Failure, Nf – the number of cycles a material (or a component) can resistance fatigue failure connoted by fatigue life

• Fatigue Strength, Sf – the value of fully reversed alternating stress, σa that corresponds to a particular number of Nf

• Endurance/Fatigue Limit, Se – a stress level below which a material can be said to have infinite life

• Low Cycle Fatigue (LCF) - when Nf is in the range of 100 to 103.

• High Cycle Fatigue (HCF) - when the number of Cycles to failure, Nf , is in the range of 103 to 106.

• Infinite Life - when Nf is equal to or greater than 106.

Stress-Life Approach: FCG

• Physically, Fatigue crack growth (FCG) typically consists of crack nucleation, growth, and final fracture.

• The Stress-Life Approach measures Total Life and cannot distinguish the individual stages of FCG.

• When tracking crack length is important, the LEFM approach should be applied.

Stress-Life Approach: Basquin’s Equation

• Basquin in 1910 suggest a log-log straight line S-N relationship

B

ATri-Slope Model

Basquin Model

( )B

f fS A N=

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes.

Basquin Equation

uA S=

( ) ( )1 6 log e uB S S=

How to Find A & B Constants



• The S-N diagram is applicable for completely reversed stresses (R=-1) only!

• We must use Fatigue Failure Criterion to correct for mean stresses!

• Assumptions:– High cycle fatigue regime

– Strain range is dominated by elastic rather than plastic strain

– Peak and valley stresses are constant

– Only one component of loading (e.g., axial, torsional, bending) acting at a time

– Stress amplitude is plotted against cycles to failure (e.g. S-N diagram)

Unmodified Endurance Limit

The Endurance Limit, Se

• Def: Endurance limit, Se

the stress below which a material can be consider to have infinite life.

• The material can be subject to an infinite number of cycles without fatigue failure.

INFINITE LIFE

The Endurance Limit, Se

• For a given material the endurance/fatigue limit has an enormous range depending on:• surface finish, • size, • type of loading, • Cyclic rate (frequency),• temperature,• corrosive, and other aggressive environments, • mean stresses,• residual stresses, and • stress concentrations.

Unmodified Endurance Limit, S’e

• The unmodified Endurance limit, S’e is measured in stress amplitude, σa under completely reverse stress (R=-1) with small highly polished unnotched specimens based on 106 to 5x108 cycles to failure in laboratory air environment.

• Can range from essentially 1 to 70 percent of the ultimate tensile strength.• Example of 1 percent of Su is a high strength steel with a sharp notch

subjected to a high mean tensile stress in a very corrosive atmosphere.

• Example of 70 percent of Su is a medium strength steel in an inert atmosphere containing appreciable compressive residual stresses.

Test Specimen

• Test specimens are subjected to repeated stress while counting cycles to failure

• Most common test machine is R. R. Moore high-speed rotating-beam machine

• Subjects specimen to pure bending with no transverse shear

• As specimen rotates, stress fluctuates between equal magnitudes of tension and compression, known as completely reversed stress cycling

• Specimen is carefully machined and polished

Fig. 6–9

Test Machine

Demonstration

Unmodified Endurance Limit, S’e

• In metals, the unmodified Endurance limit, S’e is due to interstitial elements, such as carbon or nitrogen in iron which pin dislocations.

• This prevents the slip mechanism that leads to the formation of microcracks.

• Care must be taken when using the endurance limit since it can disappear due to

1. Periodic overloads (which unpin dislocations)

2. Corrosive environments (due to fatigue corrosion interaction)

3. High temperature (which mobilize dislocations)

The Endurance Limit, Se: Steel

Unmodified Endurance Limit, S’e: Nonferrous Alloys

• Nonferrous metals often do not have an endurance limit.

• Fatigue strength Sf is reported at a specific number of cycles

Plot shows typical S-N diagram for aluminums

Unmodified Endurance Limit, S’e: Polymers

Relationship of Se and SuSteel Wrought & Cast Iron

Aluminum Alloys Copper Alloys

Estimates for Steel

• Steels data clustered near the fatigue ratio Se/Su≈0.5 for the low- and medium-strength steels. The data however, actually falls between 0.35 and 0.6 for Su <1400MPa. For Su > 1400 MPa, Se does not increase significantly.

• For unnotched, highly polished, small bending specimen fatigue limits for steels are.

• The UTS can be approximated using Brinell Hardness

( )

( )

0.5 for 1400 MPa 200ksi

700 MPa for 1400 MPa 200ksi

e u u

e u

S S S

S S

3.45 for MPa units

0.5 for ksi units

u

u

S HB

S HB

Unmodified Endurance Limit, S’e: Limits

Table of Properties

• Representative monotonic tensile properties and bending fatigue limits of selected engineering alloys obtained under the above conditions are given in Table A.1.

• The fatigue limits given in Table A.1 must be substantially reduced in most cases before they can be used in design situations.

• For example, 10 to 25 percent reductions in these values for just size effect alone is not unreasonable for bend specimens greater than 10 mm (0.4 in.) in diameter.

Endurance Limit Modifiers


• The unmodified endurance limit, S’e, is based on the rotating bending tests

• The modified endurance limit, Se is based on the actual service conditions of your parts.

• Marin factors, Ci are used to account for service conditions.

• They are unitless, positive, and have a maximum value of unity.

Type of Load, CLoad

• Accounts for changes in endurance limit for different types of fatigue loading.

• Only to be used for single load types. Use Combination Loading method when more than one load type is present.

• Bending is a more favorable loading condition than axial loading because (surface initiating) fatigue cracks propagate into lower stress environments.

Size Factor, CSize

Size Factor, CSize

• Larger parts have greater surface area at high stress levels

• Likelihood of crack initiation is higher

• Size factor is obtained from experimental data with wide scatter

• For round and rotating specimen under bending and torsion loads, the trend of the size factor data is given by

• For axial load, there is no size effect, so Csize = 1


Csize

Size Factor, CSize

• For parts that are not round and rotating, an equivalent round rotating diameter is obtained.• Equate the volume of material stressed at and above 95% of the maximum

stress to the same volume in the rotating-beam specimen.• Lengths cancel, so equate the areas.

• For a rotating round section, the 95% stress area is the area of a ring,

• Equate 95% stress area for other conditions to Eq. (6–22) and solve for d as the equivalent round rotating diameter


Size Factor, CSize

• For non-rotating round,

• Equating to Eq. (6-22) and solving for equivalent diameter,

• Similarly, for rectangular sections, h x b, A95 = 0.05 hb. Equating to Eq. (6–22),

• Other common cross sections are given in Table 6–3


Size Factor, CSize

Table 6–3

A95 for common non-rotating structural shapes


Example 6–4


Csize

Csize

Csize

Surface Condition Factor, CSurf

• Unmodified endurance limit determined from highly polished samples

• Data is typically quite scattered

• Small variation of surface leads to large variation of life



Example 6–3


CSurf

CSurf

Temperature Factor, CTemp

• At high temperatures, endurance limit declines compared to RT

• Equations based on steel (not for Al, Mg, and Cu alloys)

Reliability Factor, CR

• S-N data has a statistical nature

• Most endurance data reported as mean values

• Reliability = 1 - Probability of Failure


Reliability Factor, CR

• Simply obtain CR for desired reliability from the table below

Table 6–5

CR


Environment, CEnv

• Note: Chemical attack can create pits at which fatigue cracks start.

• Corrosion can greatly reduce the portion of fatigue life devoted to fatigue crack initiation and growth and thus greatly reduce the fatigue strength at long lives.

Miscellaneous Factor, CMisc

• Quantity designated for various factors that are not easy to account for:• Residual stresses• Directional characteristics from cold working• Case hardening• Corrosion• Surface conditioning, e.g. electrolytic plating and metal spraying• Cyclic Frequency• Frettage Corrosion• Radiation effects

• In these situations, failure is likely to be more accelerated and the endurance limit lower than expected from separate estimates

• The value of CMisc should be lower near stress concentrations


• Once you have identified all the modifiers multiply them together to calculate the modified endurance limit, Se which approximates the real endurance limit expected in your component under service conditions.

Mean Stress

Mean Stress

Mean Stress

• Tensile and/or compressive mean loads and fully reversed loads are prevalent in all fields of engineering.

• Examples of different mean loadings.• The transmission history indicates

significant tensile mean stress.• The suspension history shows

significant compressive mean stress loading.

• The bracket history is dominated by essentially fully reversed, R= -1, loading.

Mean Stress

• A thin or thick-walled pressure vessel subjected to cyclic internal pressure represents a component subjected to mean tensile stresses.

• Helical compression springs are actually undertorsion, but the applied cyclic forces involve compressive mean forces.

• A cantilever beam deflected at the free end and then released to vibrate represents a damped vibration with essentially zero mean stress.

Mean Stress

min

max

R

= max min

2m

+=

Fully Reversed: R = -1; A = ∞Zero to Tension: R = 0; A = 1Min equal to Max: R = 1; A = 0

Zero Mean Stress

Increasing Mean Stress

The mean stress, Sm, can have substantial influence on fatigue behavior.

In general, tensile mean stresses are detrimental and compressive mean stresses are beneficial.

Mean Stress

• At intermediate or high stress levels under load control test conditions, substantial cyclic creep (also referred to as cyclic ratcheting) which increases the mean strain, can occur in the presence of mean stresses.

• This cyclic creep adds to the detrimental effects of tensile mean stress on fatigue life and results in additional undesirable excess deformation.

Mean Stress

• There exist many methods to plotting the relationship between alternating stress, Sa and mean stress, Sm

• S-N curve generated from fully reversed, (Sm= 0, R = -1), fatigue limit of smooth specimens

• Similar behavior exists for other alloys.

• The general trend indicates that tensile mean stresses are detrimental.

• Much of the data fall between the straight and curved lines.

Master Fatigue Diagram

• Displays four stress components and two stress ratios

• Maximum Stress

• Minimum Stress

• Alternating Stress

• Mean Stress

• Stress Ratio

• Amplitude Ratio

Endurance Limit CurveFinite Life Curves




Master Fatigue Diagram

• Displays four stress components and two stress ratios

• Maximum Stress

• Minimum Stress

• Alternating Stress

• Mean Stress

• Stress Ratio

• Amplitude Ratio

SAFE

NOT SAFE

Example: Infinite Life Design

Goodman or Haigh Diagram

a m

a

m

Goodman or Haigh Diagram

INFINITELIFE

FINITELIFE

a m

a

m

Plot of Alternating vs Midrange Stress• Experimental data on normalized plot of a vs m

• Demonstrates little effect of negative midrange stress

Fig. 6–25

Add Tensile Mean Stress

Endurance Strength Decreases

Mean Stress

Tensile Mean Stress Influence on Long-life (5x10^7Cycles) Fatigue Strength For Aluminum Alloys

Add Tensile Mean Stress

Endurance Strength Decreases

Fatigue Failure Criteria

• Gerber passes through the data

• ASME-elliptic passes through data and incorporates rough yielding check

• Modified Goodman is linear, so simple to use for design. It is more conservative than Gerber.

• Soderberg provides a very conservative single check of both fatigue and yielding. Rarely Used.

• Langer line represents standard yield check. It is equivalent to comparing maximum stress to yield strength.



• To determine factor of safety against fatigue failure, replace stress amplitude and mean stress with nfσa and nfσm, respectively; solve for nf

• Modified Goodman

• Gerber

• ASME-Elliptic and Langer

How to Use Fatigue Criteria

• Find Cycles to Failure given Mean Stress

• If we have fatigue with mean stress, Sm we need to use a fatigue criteria todetermine the equivalent fully-reversed alternating stress, SNf condition.The SNf is found by replacing Se with SNf.

• Determine the onset of Yielding

• Check for yielding by replace Su with the monotonic yield strength Sy. And Se with the cyclic yield strength SNf.

• No yielding if <1

1

1

a m

e ut

a m

Nf ut

S S

S S

+ =

+ =

Example: Modified Goodman

1

1

a m

e ut

a m

y y

S S

S S

+ =

+ =

Summarizing Tables for Failure Criteria

• Tables 6–6 to 6–8 summarize the pertinent equations for Modified Goodman, Gerber, ASME-elliptic, and Langer failure criteria

• The first row gives fatigue criterion

• The second row gives yield criterion

• The third row gives the intersection of static and fatigue criteria

• The fourth row gives the equation for fatigue factor of safety

• The first column gives the intersecting equations

• The second column gives the coordinates of the intersection

Summarizing Table for Modified Goodman

Summarizing Table for Gerber

Summarizing Table for ASME-Elliptic

Solving Problems

Using the Book

• In our Book, Basquin’s equation using σf' is based on reversals, 2Nf, rather than cycles, Nf.

• Sa or SNf= σf' (2Nf)^b

• Values of the σf', and b, defined as the fatigue strength coefficient and the fatigue strength exponent respectively, are given in Table A.2 for a few selected engineering alloys.

• When doing your HW you will need to keep this in mind!

Using the Book

• The slope, B, depends upon many factors and for unnotched parts could vary from about –0.05 to –0.2.

• For example, the varying slope can be indicated by using surface effects with Sf taken from Fig. 4.15.

• Surface effects are dominant at long fatigue lives and less significant at short lives with convergence of the S-N curves at Suat Nf= 1.

• This gives different values of slope B for each surface condition.

• The convergence at Nf= 1 is reasonable because surface finish does not have an appreciable affect on monotonic properties for most smooth metal specimens.

Using the Book

• Oftentimes the slope, B, for smooth unnotched specimens is about -0.1.• This suggests that for unnotched specimens the fatigue life is approximately

inversely proportional to the 10thpower of alternating stress.

• Thus, a 10% increase or decrease in alternating stress will cause about a factor of three decrease or increase, respectively, in fatigue life.

• For notched parts the slope of the S-N curve on logarithmic scales is steeper yielding more extreme changes.

• Thus, even small changes in applied alternating stress can have a significant effect on fatigue life.

How to Plot S-N Curve without Data

• Sometimes you need to approximate the S-N Curve when only the Su and/or Seare known!

• Basquin Approach• Simple

• More Conservative

• Tri-Slope Approach• Slightly more complex

• More Accurate

B

ATri-Slope Model

Basquin Model

Basquin Approach

• Basquin uses the Ultimate Tensile Strength as a reference point.

B

A

Basquin Model

( )B

f fS A N=

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes

Basquin Equation

uA S=

( ) ( )1 6 log e uB S S=


Tri-Slope Approach (for HCF)

• Tri-Slope uses the S1000 point corresponding to 1000 cycles as a reference point

B

ATri-Slope Model ( )

B

f fS A N=

Note: the Tri-Slope Model can be used when a distinct bend in observed at the transition point between the LCF and HCF regimes

Basquin Equation

( )2

1000

e

SA

S= ( ) ( )10001 3 log eB S S= −


( )3

100010 , S

Tri-Slope Approach (for S1000)

Generally only modification factors for S1000 are load type and reliability

( )3

100010 , S

1000 load US C S=

Cload

S1000

Finite Life• We can make finite life predictions by replacing the endurance limit,

Se with a finite life such as S10,000.

10,000S

Summary➢ The Stress-Life method does not work well in low-cycle fatigue, where the applied strain

have a significant plastic component.

➢ The endurance limit under constant amplitude loading conditions occurs for a few metals (notably low and medium strength steels), but under in-service variable amplitude loading with corrosive, temperature, or other environmental conditions, the fatigue limit is rare.

➢ Consider the Marin Factors and note, they do not account for interactions.

➢ Yielding, finite or long life, and mean stress effects can be approximated using the five-fatigue failure criterion.

➢ Don't neglect the advantages of compressive mean or compressive residual stresses in improving fatigue life and the detrimental effect of tensile mean or tensile residual stresses in decreasing fatigue life.

➢ Do attempt to use actual fatigue data in design, but if this is not possible or reasonable, approximate estimates of median fatigue behavior can be made.

Summary➢ Consider a critical location in component

➢ Consider the mechanical (and thermal) loads experienced by that component

➢ Use Solid Mechanics analysis or FEA to convert external loads to internal stresses

➢ Determine the loading rate and the cyclic stress history

Ex: Hip prosthesis

(generic Ti alloy)

Homework 4

• Problem 4.1 from the text.



• Problem 4.11 from the text

References

• Stephens, R.I., Fatemi, A., Stephens, R.R., Fuchs, H.O., 2000, Metal Fatigue in Engineering 2nd Edition, Wiley.

• Suresh, S., 1998, Fatigue of Materials 2nd Edition, Cambridge University Press.

• Bannantine, J., 1989, Fundamentals of Metal Fatigue Analysis, Pearson.

• Schijve, J., 2008, Fatigue of Structure and Materials 2nd Edition, Springer.

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

[email protected]

me.utep.edu/cmstewart/

mailto:[email protected]

http://me.utep.edu/cmstewart/

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Fatigue Analysis - UTEP