Design for Variable Loading 2 2012

Design for Variable Loading It has been established by experiment that components fail when loads area repeated and reversed 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 point of failure, and is particularly dangerous for components with discontinuities since these always produce points of stress concentration.

Lecture Content

• Significance of the Endurance Limit. • Endurance Limit –  Limit –  Modifying  Modifying Factors. • Graphical determination of fatigue strength under fluctuating lad conditions.

Lecture Content

• Significance of the Endurance Limit. • Endurance Limit –  Limit –  Modifying  Modifying Factors. • Graphical determination of fatigue strength under fluctuating lad conditions.

Solution  –  Results  Results Summary Ultimate Tensile Strength

=

950 MPa

Endurance Limit (based on

=

257 MPa

=

127 MPa

material/loading condition) Modified Endurance Limit (for actual conditions)

Graphical Determination of Fatigue Strength Under Fluctuating Load • In practice the cyclic stress applied to an element may be considered as a combination of an alternating stress superimposed on a constantly applied mean stress. Stress Amplitude,  a

Stress

Mean Stress,  m

Stress Range,

( 2  a )

r  

  

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 maintained.

Modified Goodman Diagram Known parameters:

Ultimate tensile strength, Su Yield Strength, Sy

Alternating Stress,   a S u

Modified endurance limit, Se

S  y

S e

S  y 

S e

S u

Mean Stress,  m

Modified Goodman Diagram Known parameters:

Ultimate tensile strength, Su Yield Strength, Sy

Alternating Stress,   a S u

Modified endurance limit, Se

S  y

S e

S  y 

S e

S u

Mean Stress,  m

Modified Goodman Diagram Known parameters:

Ultimate tensile strength, Su Yield Strength, Sy

Alternating Stress,   a S u

Modified endurance limit, Se

S  y

S e

S  y 

S e

S u

Mean Stress,  m

Modified Goodman Diagram Known parameters:

Ultimate tensile strength, Su Yield Strength, Sy

Alternating Stress,   a S u

Modified endurance limit, Se

S  y

S e

S  y 

S e

S u

Mean Stress,  m

Modified Goodman Diagram Known parameters:

Ultimate tensile strength, Su Yield Strength, Sy

Alternating Stress,   a S u

Modified endurance limit, Se

S  y   

amax

S e

  

amax

  

mmax



S e

S  y

S u

Mean Stress,  m

Complete Modified Goodman Diagram Known parameters: Alternating Stress,   a

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e

S u S  yt 

S e S  yc

S  yt  

S e

S  yc

S u

Mean Stress,  m

Complete Modified Goodman Diagram Known parameters: Alternating Stress,   a

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e

S u S  yt 

S e S  yc

S  yt 



S e

S  yc

S u

Mean Stress,  m

Complete Modified Goodman Diagram Known parameters: Alternating Stress,   a

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e

S u S  yt 

S e S  yc

S  yt 



S e

S  yc

S u

Mean Stress,  m

The diagram can be simplified considering the symmetry about the diagonal axis and by rotating it through 45° .

Complete Simplified Goodman Diagram Known parameters:

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e Stress Amplitude,  a

S  y

S e

S uc

S  yc

S yt 

S ut 

Mean Stress,  m

Complete Simplified Goodman Diagram Known parameters:

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e Stress Amplitude,  a

S  y

S e

S uc

S  yc

S yt 

S ut 

Mean Stress,  m

Complete Simplified Goodman Diagram Known parameters:

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e Stress Amplitude,  a

S  y

S e

S uc

S  yc

S yt 

S ut 

Mean Stress,  m

Complete Modified Goodman Diagram Known parameters: Alternating Stress,   a

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e

S u S  yt 

S e S  yc

S  yt 



S e

S  yc

S u

Mean Stress,  m

Complete Simplified Goodman Diagram Known parameters:

UTS, Su;Yield Strength, Sy ;Mod. End. limit, S e Stress Amplitude,  a

S  y

  

S e

a

  

m

  

amax

S uc

S yt 

S  yc   

mmax

S ut 

Mean Stress,  m

Should either the yield strength, or the ultimate tensile strength, be unobtainable, a further simplification can be made.

Simplified Goodman Diagram Known parameters:

UTS, Su or Yield Strength, Sy ;Mod. End. limit, S e

Stress Amplitude,  a

Modified Goodman Line

S e

Soderberg Line

S uc

S  yc

S yt 

S ut 

Mean Stress,  m

Design for Variable Loading  –  Worked Example

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 a corresponding fillet radius of 5 mm. The bar should have an infinite life and is subject to a factor of safety of 2.

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

10 7

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

Solution

Sut = 430 MPa Un-modified endurance limit for reversed torsion:

S e

'



0.55(430)  236 MPa

Solution - Surface Finish, k a Ka=0.68 Surface factor, k a

1.0 0.8

X Polished Ground Machined/Cold Drawn

0.6 0.4 0.2

Hot Rolled Forged

0

0.5

1.0 1.5 Tensile strength, S  (GPa)

Solution  –  Size Effect, k b

For Axial Loading: 5



S 'e  0.566  9.68 x10 S uc S uc

Assuming S uc S e

'



S ut 



0.566



9.68 x10

5



430430  225MPa

This is the ‘book value’ endurance limit including size factor.

Solution - Stress Concentration, k e • Applicable to both ductile and brittle materials when subject to fatigue loading. k e

• Where



1

1  q K t   1

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)

Notch Sensitivity Chart For Steel and Aluminium Alloys  Notch radius = 5 mm. Extrapolate to determine suitable value for q 1.0

X

0.8

 Notch

0.6

Steel: Sut = 1.4GPa Sut = 1.0GPa

Sensitivity 0.4

Sut = 0.7GPa

q 0.2

Sut = 0.4GPa Aluminium Alloy

0

1.0

2.0  Notch Radius, r (mm)

3.0

4.0

Solution - Stress Concentration, k e k e



1 1  q( K t   1)



1 1  0.8(2  1)



0.56

Solution  –  Other Factors All other modifying factors are assumed to have no effect and hence equal unity.

k c



k d 



k  f    1

Solution  –  Modified Endurance Limit, Se

S e



'

S e k a k c



225 x0.56 x0.68  86 MPa

 Remember S e’ already includes a size factor, k b

Solution  –  Applied Stress Stress range

Stress amplitude

Mean stress

Static stress

Solution  –  Applied Stress

Static Stress

  

 sta tic



 F  s  A

3

3

50 x10 

  d 

63.7 x10 

2

d 

2

4

Stress Range

range 

  

 F range  A



amplitude

range



2

2

d 

  



d 2

mean     static    amplitude 

  

amplitude

  

mean



0.5

d 2

3

Mean Stress   



127.3 x103

4

63.7 x10

     

100 x103

127.3 x103 d 2

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.

Stress Amplitude, MPa

Solution  –  Goodman Diagram

S  y (215)

S e (86)

S  y (215)

S ut  (430) Mean Stress, MPa

Stress Amplitude, MPa

Solution  –  Goodman Diagram

S  y (215)

S e (86)

S  y (215)

S ut  (430) Mean Stress, MPa

Stress Amplitude, MPa

Solution  –  Goodman Diagram

S  y (215)

S e (86)

‘Safe’ S  y (215)

S ut  (430) Mean Stress, MPa

Stress Amplitude, MPa

Solution  –  Goodman Diagram

S  y (215)

  

a



0.5

  

m

S e (86)

S  y (215)

S ut  (430) Mean Stress, MPa

Stress Amplitude, MPa

Solution  –  Goodman Diagram

S  y (215) acritial  

  

63 MPa;  mcritical 

  

a





125 MPa

0.5

  

m

S e (86)   

acritial 

  

mcritial 

S  y (215)

S ut  (430) Mean Stress, MPa

Solution From the Goodman Diagram:

mcritical  

  

125 MPa 125

Including Factor of Safety:

  

mcritical 





62.5 MPa

2

Relating to strength calculation: mcritical  

  



d 2

127.3 x103



2

d 

127.3 x103 62.5 x10

6



62.5 MPa

, d   45.1mm

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.