Machine Design II Prof. K.Gopinath & Prof. M.M.

Mayuram

Our real concern is how to design a component so that failure by fatigue could be precluded.

We have noted earlier that

-Materials response to fatigue loading is characterized by its S-N behavior obtained through

a standard test

-The most important factors that affect the fatigue performance (strength) are also noted in

the previous lecture.

-Standard test conditions do not account for all these factors.

-Components in real use will be subjected to different or varied conditions.

In order to design for satisfactory fatigue life (prior to testing actual components), good

practice requires that the "laboratory" Endurance Limit value be reduced by several

adjustment factors. These reductions are necessary to account for:

(a) the differences between the application and the testing environments, and

(b) the known statistical variations of the material.

This procedure is to insure that both the known and the unpredictable factors in the

application (including surface condition, actual load, actual temperature, tolerances,

impurities, alloy variations, heat-treatment variations, stress concentrations, etc. etc. etc.)

will not reduce the life of a part below the required value. Please read that paragraph again,

and understand it well.

An accepted contemporary practice to estimate the maximum fatigue loading which a

specific design can survive is the Marin method, in which the laboratory test-determined EL

of the particular material (tested on optimized samples) is adjusted to estimate the

maximum cyclic stress a particular part can survive.

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

This adjustment of the EL is the result of six fractional factors. Each of these six factors is

calculated from known data which describe the influence of a specific condition on fatigue

life.

Those factors are:

(a) Surface Condition (ka): such as: polished, ground, machined, as-forged, corroded, etc.

Surface is perhaps the most important influence on fatigue life;

(b) Size (kb): This factor accounts for changes which occur when the actual size of the part

or the cross-section differs from that of the test specimens;

(c) Load (Kc): This factor accounts for differences in loading (bending, axial, torsional)

between the actual part and the test specimens;

(d) Temperature (kd): This factor accounts for reductions in fatigue life which occur when

the operating temperature of the part differs from room temperature (the testing

temperature);

(e) Reliability (ke): This factor accounts for the scatter of test data. For example, an 8%

standard deviation in the test data requires a ke value of 0.868 for 95% reliability, and 0.753

for 99.9% reliability.

(f) Miscellaneous (Kf): This factor accounts for reductions from all other effects, including

residual stresses, corrosion, plating, metal spraying, fretting, and others.

These six fractional factors are applied to the laboratory value of the material endurance

limit to determine the allowable cyclic stress for an actual part: Real-World Allowable

Cyclic Stress = ka * kb * Kc * kd * ke * kf * EL

Thus designers are now able to tackle this situation by applying as many modification

factors as possible so that most important deviations of the real design condition from the

standard test conditions are accounted. So the next part of the discussion will deal with the

endurance strength modification factors.

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

Endurance Strength Modification Factors

The most important deviations that occur in design situation compared to standard test

conditions are

• Size variations

• Surface finish differences

• load variations

• temperature differences

• Other miscellaneous-effects

Differences-load variations-temperature differences-other miscellaneous-effects To account

for these conditions a variety of modifying factors, each of which is intended to account for

a single effect, is applied to the endurance limit value of test specimen obtained under

laboratory conditions. Consequently we may write

Se = Se* ka kb kc kd keS = endurance limit of mechanical element (to be designed) Se* =

endurance limit of test specimen. ka = surface factor kb = size factor kc = load factor kd =

temperature factor ke = miscellaneous-effects factor

Modification Factors

Surface Factor ka

the surface of the rotating-beam specimen is highly polished, with final polishing in the axial

direction to smooth out any circumferential scratches. For other conditions the modification

factor depends upon the quality of the finish and upon the tensile strength. Sufficient data

is available in the literature relating the basic strength of the material and its surface finish

or surface condition to the modification factor which is nothing but the percentage of

standard endurance that could be realized under this condition. Typical charts are given

below. A more practical approach can be to use an empirical relation of the type ka = aSbut

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

is available in literature to account for the various surface condition values of constant a

and b are shown in the table below.

FACTOR a
SURFACE FINISH EXPONENT b
Kpsi MPa

Ground 1.34 1.58 -0.085

Meachined or cold rolled 2.70 4.51 -0.265

Hot Rolled 14.4 57.7 -0.718

As- forged 39.9 272 -0.995

100
Polished
90
Ground
80

Machined
70

60

50
Hot rolled
40
As Forge
30
corroded in
tap water
20
corroded in salt water
10

0
300 600 800 1000 1200 1400 1600 1800
Tensile Strength (MPa)

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

1.1
1.6
1.0 0.025
0.05
0.9 0.1
0.2
0.4
0.8 1.6 0.8
6.3 3.125
0.7 12.5
25.0
0.6 50.0

0.5
0.4
560 840 1120 1400 1680
280
Failute Strength Sut (MPa)

Size Factor Kb

The size factor accounts for the variations in the size of the component when

compared to the test specimen. The size factor has been evaluated using sets of data

points, from available literature. The larger the size higher the probability of internal defects,

hence lower the fatigue strength. An empirical relation for the case of bending and torsion

can be expressed as given below

⎧⎪ d / 7.62 )−0.107 = 1.24d −0.107 2.79 ≤ d ≤ 51mm ⎫⎪

k b = ⎨( ⎬
⎩⎪0.859 − 0.000837d 51 ≤ d ≤ 254mm ⎭⎪

Size Factor

For large sizes, kb further reduces to 0.60 and lower Note that for axial loading there is

no size effect, therefore use kb = +1.0 in this case

Load Factor-Axial Loading

Though there is no apparent size effect for specimens tested in axial or push–pull

fatigue, there is definite difference between the axial fatigue limit and that in reserved

bending. A very extensive collection of data has been made by R.W.Landgraf (Ford motor

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

company), on axial fatigue. These results were analyzed, resulting in the modification

values for axial loading as shown in the table.

Load Factor-Torsional Loading

A collection of 52 data points comparing the torsional endurance limit with the bending

endurance limit yielded a load factor for torsion of 0.565. Using a different set of data points,

Mischke obtained the result kc =0.585. Both of these are very close to the value of 0.577

shown in the table. Note that this value incidentally happen to be the relation between

torsional and tensile yield strengths according to the distortion energy theory.

Load Factor

Hence for the three basic types of loading normally encountered in most practical

applications, namely axial, bending and torsional stressing the effect could be accounted by

the load factor as shown in the table

⎧0.923 Axial Loading S<1520MPa(220Kpsi) ⎫

⎪1 S<1520MPa(220Kpsi) ⎪⎪
⎪ Axial Loading
kc = ⎨ ⎬
⎪1 Bending ⎪
⎩⎪0.577 Torsion and shear ⎭⎪

Temperature factor

The limited amount of data available show that the endurance limit for steels in creases

slightly as the temperature rises and then begins to fall off in the 400 to 700 ˚F range, not

unlike the behaviour of the tensile strength shown figure below

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

1.0
Sut
0.9

0.8 Sy
ST/SRT
0.7

0.6

0.5
0 RT
200 400 600

Temperature, oC

For this reason it is probably true that the endurance limit is related to tensile strength at

clevated temperatures in the same manner as at room temperature. It seems quite logical,

therefore, to employ the same relations to predict endurance limit at elevated temperatures

as are used at room temperature, at least, this practice will provide a useful standard

against which the performance of various materials can be compared.

Two types of problems arise when temperature is a consideration. If the rotating- beam

endurance limit is known at room temperature, then use

ST
kd =
SRT

Miscellaneous – Effects Factor Ke

Similarly the other factors take into account the deviations of actual condition of use

from the standard testing. Though the factor ke is intended to account for the reduction in

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

endurance limit due to all other effects, it is really intended as a reminder that these must

be accounted for, because actual conditions of use vary from standard test condition;

values of ke are not always available.

Endurance - Limit - Preliminary Observations

The determination of endurance limit by fatigue testing is now a routine, though a

lengthy procedure. Generally stress testing is preferred to strain testing for endurance limits.

For preliminary and prototype design and for some failure analysis as well, a quick method

of estimating endurance limit is needed. There are great quantities of data in the literature

on the results of rotating-beam tests and simple (static) tension tests of specimen taken

from the same bar or in got. By plotting the resulting tensile and endurance strength values

as in shown in Figure, it is possible to see whether there is any correlation between the sets

of results. The graph appears to suggest that the endurance limit ranges from about 40 to

60 percent of the tensile strength for steels up to about 1400 MPa (200 kpsi). Beginning at

about Sut = 1400 MPa (200 kpsi), the scatter appears to increase, but the trend seems to

level off, as suggested by the dashed horizontal line at S’e=700MPa (100 kpsi).

Hence for preliminary design purposes the standard laboratory endurance strength of can

be derived from its ultimate tensile strength values using the following relations

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

Stress Concentration Effect

Fatigue Stress Concentration

The existence of irregularities or discontinuities, such as holes, grooves, or

notches, in a part increase the magnitude of stresses significantly in the immediate vicinity

of the discontinuity. Fatigue failure mostly originates from such places. Hence its effect

must be accounted and normally a fatigue stress-concentration factor Kf is applied when

designing against fatigue, even if the materials behavior is ductile.

Fatigue Stress Concentration Factor

Recall that a stress concentration factor need not be used with ductile materials

when they are subjected to only static loads, because (local) yielding will relieve the stress

concentration. However under fatigue loading, the response of material may not be

adequate to nullify the effect and hence has to be accounted. The factor Kf commonly

called a fatigue stress concentration factor is used for this. Normally, this factor is used to

indicate the increase in the stress; hence this factor is defined in the following manner.

Fatigue stress concentration factor can be defined as

Fatigue strength (limit) of unnotched specimen

k =
f
Fatigue strength (limit) of notched free specimen

The other form of use, where necessary is the miscellaneous-effects factor ke applied as a
strength reduction factor on the fatigue limit value. With this approach we define

1
ke =
Kf

Indian Institute of Technology Madras

Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram

980 Carbon Steel 140

840
Alloy Steel 120
+ Wrought irons
700 100 Kpsi

560 80

420 60

280 40
+ +++
140 +++ 20

0 0
0 140 280 420 560 700 840 980 1120 1260 1400 1540 1680 1820 1960 2100

Tensile Strength Sut ,MPa

Indian Institute of Technology Madras