High Cycle Fatigue (HCF) analysis
(last updated 2011-09-27)
Kjell Simonsson
1
Aim
For small loadings/long lives (with respect to the number of load cycles),
fatigue life calculations are generally stress-based.
The aim of this presentation is to give a short introduction to stressbased High Cycle Fatigue (HCF) analysis.
Kjell Simonsson
2
Basic observations; Wöhler-diagrams/SN-curves
Experiments made on smooth test specimens result in stress-life curves (so
called SN-curves) of the type shown below (such diagrams are also
referred to as Wöhler-diagrams)
log  a
log 
log 
 max   min
a 
'
f

fl
N
f
2
Fatigue limit (or Endurance limit),
“utmattningsgräns” in Swedish
No. of cycles to failure
2N
No. of load reversals to failure
f
fl
log 2 N
f
In the region of longer lives, the curve may often be described by the
Basquin’s relation
following relation
log  a  log 
'
f
 b log 2 N
  log  a
f
 log 
'
f
2 N f b   a

'
f
2 N f b
For some materials a true fatigue limit exists, while for others it is defined
as the stress for which the material can withstand a certain (large) number
of load cycles, say 1E6 or 1E7.
3
Kjell Simonsson
Wöhler-diagrams/SN-curves; cont.
Note that
• The fatigue limit could for a mild steel be in the order of 140 MPa
• There is often a large scatter in fatigue data; thus the SN-curve
represents the mean behavior
• Aluminum is a material that lacks a fatigue limit
• Basquin’s relation is sometimes rewritten in the following way
a
1/b
Kjell Simonsson
 
'
f

1/b
2N
 a N
m
f
f
 K ,m  
1
b
4
& K 
1
2 
'
f

1/b
Mean stress effects
Experiments reveal that the mean stress has a marked effect on the fatigue
life of uniaxially loaded specimens, while it has a much weaker influence on
specimens loaded in pure torsion. Basically, a positive mean stress
decreases the fatigue life. In what follows we will concentrate on the case of
a positive mean stress.
In order to fully describe this kind of behavior, we note that we are to deal
with a relation between three quantities, namely the stress amplitude σa ,
the mean stress level σm and the fatigue life Nf , which for instance may be
written
N f  f  a ,  m 
However, such a relation will (for each material in each specific condition)
require an enormous amount of test data to set up. Now, since we when
adopting a so called total life approach (not distinguishing between different
stages of the fatigue process) generally are interested in infinite life, the life
variable becomes a parameter (a chosen large value, 1E6 or 1E7). Thus
N   f  a ,  m 
Kjell Simonsson

 a  g  m 
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Mean stress effects; cont.
Mean stress effects; cont.
From the previous slide we have
 a  g  m 
The most widely used relations of this type are
Soderberg’s relation  a


  fl 1   m
Y





  m 

 uts  
Goodman’s relation  a   fl 1  



 m  
  fl 1  
 
  uts  

2
Gerber’s relation
Kjell Simonsson
a
6
One may note that all three of
the relations to the left can be
cast in the following form

 m  
  fl 1  
 
  0  

m
a
where σ0 is a reference
stress (the yield stress
or the ultimate tensile
stress) while m is a
parameter.
Mean stress effects; cont.
Mean stress effects; cont.
For a mild steel we get the following form for the Soderberg, Goodman and
Gerber relations, resp.

160 a

fl
 MPa 
140
Soderberg’s relation  a


  fl 1   m
Y

Goodman’s relation  a


  fl 1   m
  uts

120
100
80
60







 m  
  fl 1  
 
  uts  

2
40
Gerber’s relation
20
0
0
100
200
Y
300
400
 uts
500
 m  MPa

a
Experiments seem to indicate that the true response typically can be found
in between the Goodman and the Gerber curves, and that the usage of the
former thus provides a conservative approach.
Kjell Simonsson
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The Haigh-diagram
Instead of proposing one single function describing the amplitude stress as a
function of the mean stress (for infinite life) as done above, one may of
course use a multi-linear function. One such example is the so called
Haigh-diagram, see below.


160
fl
flp
 a  MPa

Data for the same
mild steel as
discussed above.
140
120
N
100
f

80
60
40
20
0
0
100

200
flp
300
400
500
 uts
 m  MPa

In addition to the fatigue/endurance limit σfl on the ordinate (for an
alternating loading), and the ultimate tensile strength σuts on the abscissa
(for a static loading), one also uses experimental data for a pulsating
loading.
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Kjell Simonsson
The Haigh-diagram; cont.
Note that
• No distinction is made between different fatigue stages (i.e. crack
initiation and crack propagation) in the Wöhler- and Haigh-diagrams.
Therefore, we refer to analyses based on these as examples of
total life approaches to fatigue design.
• The Haigh-diagram allows, in its standard form, only design with respect
to infinite life.
• Values for σfl and σflp are generally recorded for uniaxial loading,
bending and torsional loading.
• The reason why only two combinations of σa and σm are used in the
Haigh-diagram is the economical- and time cost of performing fatigue
experiments.
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The Haigh-diagram; cont.
Sometimes a line representing the yield limit is included in the Haighdiagram, since plastic flow is to be avoided.
 a  MPa

300
Y


Data for the same
mild steel as
discussed above.
250
200
fl
150
N
flp
f

100
50
0
0
Kjell Simonsson
100

200
flp
Y
300
10
400
 uts
500
 m  MPa

Topics still not discussed
Topics for the next lecture
It is important to note that the material response found in material tables etc
is found for laboratory specimens tested at laboratory conditions. For
industrial/real life applications the fatigue data and/or working point must be
adjusted, and this will be one of the topics for the next lecture.
Furthermore, in real life applications the loading will most likely not be as
simple as an oscillating one. Thus, on the next lecture the most widely used
cycle count method, the so called Rain Flow Count method, will be
presented (multi-axial fatigue will briefly be discussed later on).
Kjell Simonsson
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