Design for Variable Loading 2 2012

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Design

Design for

for Variable

Variable Loading

Loading

It has been established by experiment that It has been established by experiment that

components fail when loads area

components fail when loads area repeated andrepeated and reversed several million times even though

reversed several million times even though thethe stresses involved do not reach the elastic limit of stresses involved do not reach the elastic limit of the material. Fatigue failure is characterised by an the material. Fatigue failure is characterised by an absence of elongation and of reduction at the point absence of elongation and of reduction at the point of failure, and is particularly dangerous for

of failure, and is particularly dangerous for components with discontinuities since these components with discontinuities since these always produce points of stress concentration. always produce points of stress concentration.

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Lecture Content

•• Significance of the Endurance Significance of the Endurance Limit.Limit. •• Endurance LimitEndurance Limit – – Modifying Factors. Modifying Factors.

•• Graphical determination of fatigue strength underGraphical determination of fatigue strength under fluctuating lad conditions.

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Lecture Content

•• Significance of the Endurance Significance of the Endurance Limit.Limit. •• Endurance LimitEndurance Limit – – Modifying Factors. Modifying Factors.

•• Graphical determination of fatigue strength underGraphical determination of fatigue strength under fluctuating lad conditions.

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Solution

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Results Summary

Ultimate

Ultimate Tensile Tensile Strength Strength == 950 MPa950 MPa Endurance

Endurance Limit Limit (based (based on on == 257 MPa257 MPa material/loading condition)

material/loading condition) Modified

Modified Endurance Endurance Limit Limit == 127 MPa127 MPa (for actual conditions)

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Graphical Determination of Fatigue

Strength Under Fluctuating Load

•• In practice the cyclic stress In practice the cyclic stress applied to an elementapplied to an element may be considered as a

may be considered as a combination of ancombination of an alternating

alternating stress stress superimposed superimposed on on a a constantlyconstantly applied mean stress.

applied mean stress.

Stress Stress Mean Stress, Mean Stress,  _m_m Stress Stress Amplitude, Amplitude,  _a_a Stress Range, Stress Range,   _r_r((__ 22  _a_a))

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Modified Goodman Diagram

The Modified Goodman Diagram can be constructed for any material when the ultimate tensile

strength, yield strength and endurance limit for a 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

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Modified Goodman Diagram

Known parameters: Ultimate tensile strength, S_u

Yield Strength, S_y

Modified endurance limit, S_e

u S u S y S y S e S Alternating Stress, Mean Stress, m a  e S 

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Modified Goodman Diagram

Yield Strength, S_y

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Modified Goodman Diagram

Yield Strength, S_y

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Modified Goodman Diagram

Yield Strength, S_y

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Modified Goodman Diagram

Yield Strength, S_y

u S u S y S y S e S Alternating Stress, Mean Stress, m a  e S  max m  max a  max a 

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Complete Modified Goodman Diagram

Known parameters: UTS, S_u;Yield Strength, S_y;Mod. End. limit, S_e

u S u S e S Alternating Stress, Mean Stress, m a  e S  yt S yc S yc S yt S

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Complete Modified Goodman Diagram

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Complete Modified Goodman Diagram

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The diagram can be simplified considering the

symmetry about the diagonal axis and by

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Complete Simplified Goodman Diagram

e S Stress Amplitude, Mean Stress, m a  yc S S yt uc S S ut y S

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Complete Simplified Goodman Diagram

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Complete Simplified Goodman Diagram

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Complete Modified Goodman Diagram

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Complete Simplified Goodman Diagram

e S Stress Amplitude, Mean Stress, m a  yc S S yt uc S S ut y S max a  max m  m  a 

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Should either the yield strength,

or

the

ultimate tensile strength, be unobtainable, a

further simplification can be made.

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Simplified Goodman Diagram

Known parameters: UTS, S_u or Yield Strength, S_y ;Mod. End. limit, S_e

e S Stress Amplitude, Mean Stress, m a  yc S S yt uc S S ut

Modified Goodman Line

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Design for Variable Loading

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Worked Example

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Determine the diameter of a hot drawn mild steel bar (S_ut=430MPa and S_y= 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 a corresponding 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 (S_e/S_u) 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

S_ut = 430 MPa

Un-modified endurance limit for reversed torsion:

MPa

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Solution - Surface Finish, k

_a 0 0.2 0.4 0.6 0.8 1.0 1.5 1.0 0.5

Tensile strength, S (GPa) Surface factor, k _a Polished Ground Machined/Cold Drawn Hot Rolled Forged X Ka=0.68

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Solution

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Size Effect, k

_b

For Axial Loading:

Assuming

This is the ‘book value’ endurance limit

including size factor.

ut uc

S

 uc uc e

x

S

'

0 .

566

9 .

68

10

5  

 



x



MPa

S

_e' 

0 .

566



9 .

68

10

5

430



225

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Solution - Stress Concentration, k

_e

• Applicable to both ductile and brittle materials when subject to fatigue loading.

• Where q = notch sensitivity

K _t= stress concentration factor (from charts, calculation etc.)

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







1 1



1    t e K q k

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Notch Sensitivity Chart For Steel and Aluminium Alloys 0 1.0 2.0 3.0 4.0 0.4 0.6 0.8 1.0 0.2 Steel: S_ut = 1.4GPa S_ut = 1.0GPa S_ut = 0.7GPa S_ut = 0.4GPa Aluminium Alloy Notch Sensitivity q Notch Radius, r (mm) X Notch radius = 5 mm. Extrapolate to

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Solution - Stress Concentration, k

_e 56 . 0 ) 1 2 ( 8 . 0 1 1 ) 1 ( 1 1        t e K q k

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Solution

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Other Factors

All other modifying factors are assumed to have no effect and hence equal unity.

1

   _d _f c

k

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Solution

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Modified Endurance Limit, S

_e

MPa

x

k

S

_e  _e' _a _c 

225

0 .

56

0 .

68



86

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Solution

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Applied Stress

Static stress Mean stress Stress amplitude

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Solution

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Applied Stress

Static Stress

Stress Range

Mean Stress

2 3 2 3 10 7 . 63 4 10 50 d x d x A F s sta tic      2 3 2 3 10 3 . 127 4 10 100 d x d x A F _range range      2 3 10 7 . 63 2 _d x range amplitude    2 3 10 3 . 127 d x amplitude static mean      5 . 0  mean 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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Solution

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Goodman Diagram

Mean Stress, MPa Stress Amplitude, MPa ) 86 ( e S ) 215 ( y S ) 215 ( y S S _ut(430)

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Solution

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Goodman Diagram

Mean Stress, MPa Stress Amplitude, MPa ) 86 ( e S ) 215 ( y S ) 215 ( y S S _ut(430)

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Solution

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Goodman Diagram

Mean Stress, MPa Stress Amplitude, MPa ) 86 ( e S ) 215 ( y S ) 215 ( y S S _ut(430) ‘Safe’

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Solution

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Goodman Diagram

Mean Stress, MPa Stress Amplitude, MPa ) 86 ( e S ) 215 ( y S ) 215 ( y S S _ut(430) 5 . 0  m a  

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Solution

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Goodman Diagram

Mean Stress, MPa Stress Amplitude, MPa ) 86 ( e S ) 215 ( y S ) 215 ( y S S _ut(430) 5 . 0  m a   critial m  critial a  MPa MPa critical critial m a  63 ; 125 

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Solution

From the Goodman Diagram: Including Factor of Safety:

Relating to strength calculation:

MPa critical m 125  MPa critical m 62.5 2 125    MPa d x critical m 62.5 10 3 . 127 2 3    mm d x x d , 45.1 10 5 . 62 10 3 . 127 6 3 2   

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Design For Variable Loading

15. For a design application, explain why the

endurance 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 ‘Complete Simplified Goodman Diagram. List the parameters required for each.