1

Fatigue Failure Due to Variable Loading

Section V

2

Variable Loading? What have we been ignoring? How rate the lifetime of fatigue or cyclic

loaded parts? Endurance Limit Estimating Fatigue Life Determining the Endurance Limit Characterizing Fluctuating Stress Fatigue Failure Criterion Graphically

Talking Points

3

In many actual life applications, some machine members are subjected to stresses fluctuating between levels.

Often, machine members are found to fail under the action of these repeated or fluctuated stresses.

Most careful analysis reveals that the actual maximum stresses were below the ultimate strength of the material, and quite frequently even below the yield strength.

The most distinguishing characteristic of these failures is that the stresses have been repeated a very large number of times.

This type of failure is called fatigue failure.

Variable Loading?

What have we been ignoring?

Suppose the countershaft is rotating

Static Dynamic

Is fatigue an issue? What type of stress

condition do we now have if the shaft is rotating and the loads remain in a fixed direction?

Reversed Bending

As the shaft rotates the stress alternates between Tension @ C Compression @ D

Shaft rotates 180 degrees Tension @ D Compression @ C

C

D

D

C

Reversed Bending - Fatigue Common indications

of reverse bending fatigue “Beach” Marks Dark areas indicated

in this figure are representative of abrupt or “fast” fracture

STRESS PATTERNS FORREVERSE BENDING

Unidirection Bending What does each “Beach”

mark represent? Crack slowly propagated and

then stops Illustrates how the crack

front propagates thru the cross-section

Failure in a threaded rod or bolt due to unidirectional bending

Rough area representing “fast” fracture

Common Fatigue Patterns

What type of loading caused this failure?

Fast fracture

Crack grew fromthe center outward

UNIAXIAL TENSILE LOADING

How rate the lifetime of fatigue or cyclic loaded parts?

Strain Life Ideal for low cycle fatigue

applications 1≤N≤103, where N is the

number of loading cycles Based on the plasticity at

localized regions of the part Method is typically not practical

for design use because it requires knowledge of strain concentration levels, pages 316 to 317

Fracture Mechanics Approach Requires the assumption of a

pre-existing crack Used to predict growth of the

crack with respect to a specified level of stress intensity

Pages 319 to 323 Stress Life

High fatigue life calculations 10^3≤N≤106

Large amounts of data Widely used Covered in this course

Endurance Limit Is a stress level in a material that can withstand an infinite

number of loading cycles. In your text and throughout literature on the subject, the

endurance limit is typically referenced by Se. To determine the endurance limit we use a S-N curve Always plotted on Log-Log Scale

Se

S - Strength of the material

N - Number of cycles executed

N=1 - cycle represents a load application in one direction,

removal, and then once again in the opposite direction

“Knee” of the S-N Curve

Estimating Fatigue Life Approximating fatigue

103≤N≤106

Just as we saw the linear behavior of true stress-strain when plotted on log scale, the data tends to follow a piecewise linear function.

We will use this same principal to develop a power-law for estimating points in the high cycle region on the S-N diagram.

S f aN b

6

3

6

6

3

6

3

6

10

2

1010

10

10

10

10

6

10

1000

logloglog2

gives.... )( into ngSubstituti

log3

1log

3

1

gives.... equations two thesegSubtractin

)( 610loglog

)( 31000loglog

intercept theis and slope theis where

loglog

SS

SS

SC

b

S

fS

S

Sb

CbCbS

CbCbS

Cb

CNbS

e

ut

f

e

ut

S

fS

S

Sa

2

10

2

10

6

3

C log10(a)

Finally resulting in…

Determining the Endurance Limit

A rotary device serves as an excellent means of acquiring such data in a timely manner.

Several thousand cycles can be executed rather quickly… Below is a sketch of a simple apparatus that can be

used to determine the value of the endurance limit.

Much Endurance Data on record is for steels

Mischke, one of the authors of the text has actually done an extensive study in this area and has determined that the endurance limit of the material.

Steels

It is important to note that these estimates are for clean, highly polished specimens that are free of surface defects.

S e 0.504Sut , ksi or MPa Sut 212 ksi (1460 MPa)

107 ksi Sut 212 ksi

740 MPa Sut 1460 MPa

Your text emphasizes this point by the inclusion of a prime mark above the endurance

limit symbol.

Endurance Limit (EL) Modifying Factors

Factors that can reduce the EL:

Surface condition, (ka) Size factor, (kb) Load factor, (kc) Temperature, (kd) Reliability factor, (ke) Miscellaneous-effects factor,

(kf) These factors are used to adjust

the endurance limit obtained from rotating beam specimens.

Se kakbkckd kek f S eModified EL - Marin’s Equation

Now we will discuss how to effectively estimate these modification

factors.

Surface Factor, ka

Mischke performed a regression analysis to approximate the surface factor

The surface factor, ka, takes the following form:

where Sut is the minimum tensile strength and a and b are found from the table

ka aSutb

Size Factor, kb

Once again Mischke has provided a means for estimating the EL size modification factor

The size factor arises because of the geometry of the specimen used to obtain the endurance limit

Diameter 0.30 in. Extruded or drawn bar stock

Grain elongation in the direction perpendicular to fatigue crack growth

Likelihood of surface flaws is low

kb

0.879d 0.107 0.11 d 2 in.

0.91d 0157 2 < d 10 in.

1.24d 0.107 2.79 d 51 mm.

1.51d 0.157 51 < d 254 mm.

For larger parts are more likely to contain flaws which can result in

premature material failure

For axially loaded specimens the size factor is one.

Effective circular cross-section may becomputed for non-circular geometry (see

Table 7-5.)

Loading Factor, kc

Since the usual test used to obtain the EL is the reversed bending load, modification factors are needed.

Some texts on this subject do not include this factor and require the user to implement an estimation in the EL instead.

kc 1 bending

0.85 axial

0.59 torsion

Se 0.50Su bending

0.45Su axial

0.29Su torsion

Temperature, Reliability and Miscellaneous Factors Temperature is

relatively simple to compute and understand

Reliability Factor Will not be covered in

detail in this course Extensive, through

coverage is given to this factor in the text

Statistics background is required

Miscellaneous effects Corrosion Manufacturing

process Residual stresses Coatings

All of which can have an adverse effect on the EL

kd ST

SRT

where ST and SRT are the tensile strength

at the operating and room temperatures respectively.

Characterizing Fluctuating Stress

Fatigue loading is oftentimes caused by a variable loading source.

To develop failure criterion for fluctuating stresses, which cause fatigue failures, we must characterize how the stress levels vary as time.

Sinusoidal stress oscillating about a static stress

Repeated Stress Completely reversed stress

a max min

2

m max min

2

Fatigue Failure Criterion Gerber

Modified Goodman

Soderberg

a

Se

m

Sy

1

n

a

Se

m

Sut

1

n

n a

Se

nm

Sut

2

1

Fatigue Failure Criterion Graphically