Design for Variable Loading 2 2012

8/10/2019 Design for Variable Loading 2 2012

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Design for Variable Loading

It has been established by experiment that

components fail when loads area repeated andreversed several million times even though the

stresses involved do not reach the elastic limit of

the material. Fatigue failure is characterised by an

absence of elongation and of reduction at the pointof failure, and is particularly dangerous for

components with discontinuities since these

always produce points of stress concentration.


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Lecture Content

Significance of the Endurance Limit.

Endurance LimitModifying Factors.

Graphical determination of fatigue strength under

fluctuating lad conditions.


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SolutionResults Summary

Ultimate Tensile Strength = 950 MPa

Endurance Limit (based on = 257 MPamaterial/loading condition)

Modified Endurance Limit = 127 MPa

(for actual conditions)


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Graphical Determination of Fatigue

Strength Under Fluctuating Load

In practice the cyclic stress applied to an elementmay be considered as a combination of an

alternating stress superimposed on a constantly

applied mean stress.

Stress

Mean Stress, m

StressAmplitude, a

Stress Range, )2( ar


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Modified Goodman Diagram

The Modified Goodman Diagram can be constructed

for any material when the ultimate tensile

strength, yield strengthand endurance limitfora completely reversed stress are known. It is

considered that if a variable stress is superimposed

on a steady stress, the plotted results will

determine a maximum and a minimum stress line

between which safe operating conditions can be

maintained.


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Known parameters: Ultimate tensile strength, Su

Yield Strength, Sy

Modified endurance limit, SeuS

uS

yS

yS

eS

Alternating

Stress,

Mean Stress, m

a

eS


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Yield Strength, Sy


uS

yS

yS

eS

Alternating

Stress,

Mean Stress, m

a

eS


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Yield Strength, Sy


uS

yS

yS

eS

Alternating

Stress,

Mean Stress, m

a

eS


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Yield Strength, Sy


uS

yS

yS

eS

Alternating

Stress,

Mean Stress, m

a

eS


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Yield Strength, Sy


uS

yS

yS

eS

Alternating

Stress,

Mean Stress, m

a

eS

maxm

maxa

maxa


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Complete Modified Goodman Diagram

Known parameters: UTS, Su

;Yield Strength, Sy

;Mod. End. limit, Se

uS

uS

eS

Alternating

Stress,

Mean Stress, m

a

eS

ytS

yc

S

ycS

ytS


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;Yield Strength, Sy


uS

uS

e

S

Alternating

Stress,

Mean Stress, m

a

eS

ytS

yc

S

ycS

ytS


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;Yield Strength, Sy


uS

uS

e

S

Alternating

Stress,

Mean Stress, m

a

eS

ytS

yc

S

ycS

ytS


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The diagram can be simplified considering the

symmetry about the diagonal axis and by

rotating it through 45 .


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Complete Simplified Goodman Diagram

Known parameters: UTS, Su;Yield Strength, Sy ;Mod. End. limit, Se

eS

Stress

Amplitude,

Mean Stress, m

a

ycS ytS

ucS utS

yS


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eS

Stress

Amplitude,

Mean Stress, m

a

ycS ytS

ucS utS

yS


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eS

Stress

Amplitude,

Mean Stress, m

a

ycS ytS

ucS utS

yS


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uS

uS

eS

Alternating

Stress,

Mean Stress, m

a

eS

ytS

yc

S

ycS

ytS


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eS

Stress

Amplitude,

Mean Stress, m

a

ycS ytS

ucS utS

yS

maxa

maxm

m

a


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Should either the yield strength, orthe

ultimate tensile strength, be unobtainable, a

further simplification can be made.


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Simplified Goodman Diagram

Known parameters: UTS, SuorYield Strength, Sy ;Mod. End. limit, Se

eS

Stress

Amplitude,

Mean Stress, m

a

ycS ytS

ucS utS

Modified Goodman Line

Soderberg Line


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Design for Variable Loading

Worked Example


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Determine the diameter of a hot drawn mild steel bar

(Sut=430MPa and Sy= 215 MPa) which is subject

to a tensile preload of 50 kN and a fluctuating

tensile load which varies between 0 and 100 kN.

The design of the bar ends is such that a stress

concentration factor of 2 is appropriate for acorresponding fillet radius of 5 mm. The bar

should have an infinite life and is subject to a

factor of safety of 2.


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Solution

1. Strength values from test specimen data:

Ratio (Se/Su)

Material Cycle

s

U.T.S.

(MPa)

Reversed

Bending

Reversed Axial Loading Reversed

Torsion

Mild Steel 107 380 +/-0.6 +/-0.55 +/-0.36

Medium Carbon

Steel (annealed)

107 620 +/-0.5 +/-0.45 +/-0.3

Low alloy Steel 107 950 +/-0.45 +/-0.4 +/-0.27

High Strength

Steel

107 1540 +/-0.38 +/-0.32 +/-0.2

High Strength

Alloy

108 500 +/-0.3 +/-0.24 +/-0.16


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Solution

Sut= 430 MPa

Un-modified endurance limit for reversed torsion:

MPaSe 236)430(55.0'


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SolutionSize Effect, kb

For Axial Loading:

Assuming

This is the book value endurance limitincluding size factor.

utuc SS

ucuce SSxS 51068.9566.0'

MPaxSe 2254304301068.9566.0 5'


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Solution - Stress Concentration, ke

Applicable to both ductile and brittle materialswhen subject to fatigue loading.

Where q = notch sensitivity

Kt= stress concentration factor (from

charts, calculation etc.)

(If q unknown, err on the safe side and make equal tounity)

111

t

eKqk


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Notch Sensitivity Chart For Steel and Aluminium Alloys

0 1.0 4.03.02.0

0.4

0.6

0.8

1.0

0.2

Steel: Sut= 1.4GPa

Sut= 1.0GPa

Sut= 0.7GPa

Sut= 0.4GPaAluminium Alloy

Notch

Sensitivity

q

Notch Radius, r (mm)

X

Notch radius = 5 mm. Extrapolate to

determine suitable value for q


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Solution - Stress Concentration, ke

56.0)12(8.01

1

)1(1

1

t

eKq

k


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SolutionOther Factors

All other modifying factors are assumed to have no

effect and hence equal unity.

1 fdc kkk


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SolutionModified Endurance Limit, Se

MPaxxkkSS caee 8668.056.0225'

Remember Se already includes a size factor, kb


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SolutionApplied Stress

Static stress

Mean stress

Stress amplitudeStress range


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SolutionApplied Stress

Static Stress

Stress Range

Mean Stress

2

3

2

3107.63

4

1050

d

x

d

x

A

Fs

static

2

3

2

3 103.127

4

10100

d

x

d

x

A

Frange

range

2

3107.63

2 d

xrangeamplitude

2

3103.127dx

amplitudestaticmean

5.0mean

amplitude


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Solution

Determine the limiting values of mean stress and

stress amplitude by constructing a Goodman

diagram based on the strength of the component

material and the modified endurance limit.


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SolutionGoodman Diagram

Mean Stress, MPa

Stress

Amplitude,

MPa

)86(eS

)215(yS

)215(yS )430(utS


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Mean Stress, MPa

Stress

Amplitude,

MPa

)86(eS

)215(yS

)215(yS )430(utS

Safe


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Mean Stress, MPa

Stress

Amplitude,

MPa

)86(eS

)215(yS

)215(yS )430(utS

5.0

m

a


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Mean Stress, MPa

Stress

Amplitude,

MPa

)86(eS

)215(yS

)215(yS )430(utS

5.0

m

a

critialm

critiala

MPaMPacriticalcritial ma

125;63


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Solution

From the Goodman Diagram:

Including Factor of Safety:

Relating to strength calculation:

MPacriticalm 125

MPacriticalm

5.622

125

MPad

xcriticalm

5.62103.127

2

3

mmdx

xd 1.45,

105.62

103.1276

32


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Design For Variable Loading

15. For a design application, explain why theendurance limit of the material is

modified form the book value. What factors

should be taken into account when making

this adjustment.

16. Construct (i) a Complete Modified

Goodman Diagram and (ii) a CompleteSimplified Goodman Diagram. List the

parameters required for each.


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Design For Variable Loading

17. A cylindrical component is to bemanufactured form mild steel (Sut= 430 MPaand Sy= 215 MPa) and a modified endurancelimit has been established (Se= 100 MPa). If

the component is subject to both an axialtensile preload of 50 kN and a fluctuatingload in which the ratio of the stress amplitudeto the mean stress is 0.8. Determine thediameter of the cylinder if it is subject to asafety factor of 1.5.

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Design for Variable Loading 2 2012