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1PSMN
Lecture 2P-S-M-N diagrams
What’s P-S-M-N?
• P = probability of failure
• S = stress amplitude
• M = mean stress
• N = number of cycles to failure of a ‘standard’plain fatigue test specimen
2PSMN
Stress cycle
sDsa
smin
smax
sm
Tid
spenningssyklussa
sD
smin
smax
sm - midtspenning- spenningsamplitude- maksimumsspenning- minimumsspenning- spenningsvidde
Spennin
g
( ) ( )
min max
a m
Stress ratio
Amplitude ratio
1 1
R
A R R
σ σ
σ σ
=
= = − +
S-N Curves
3PSMN
Olin Hanson Basquin (1869-1946), Northwestern University, Illinois (1910): Power law describes N vs. σσσσa [Dowling].
Principle S-N curve for smooth member under fully reversed loading
4PSMN
Nomenclature
Symbol Meaning
N number of cycles to failure, life(-time)
ND number of cycles at ‘knee-point’
n number of cycles
(σm ±) σA fatigue limit (at arbitrary mean stress, σm)
(σm ±) σAN fatigue strength (at N cycles and arbitrary mean stress, σm)
σW fatigue limit (at σm = 0)
σWN fatigue strength (at N cycles and σm = 0)
σa stress amplitude
σar equivalent fully reversed stress amplitude
σm mean stress
Basquin’s equation
( )
( )
a f A ; 1 W
f f m
Basquin's equation for the fully reversed fatigue strength
at cycles:
2
Fatigue strength coefficient ( 'true' fracture strength):
1
Fatigue strength exponent:
1
Basquin
b
N R N
N
N
R Z
b m
σ σ σ σ
σ σ
=−′= = =
≈
′ ≈ = −
= −
( )W D
1
a W D
's equation may be rewritten in terms of 'knee-point'
parameters and :
m
N
N N
σ
σ σ −=
5PSMN
Strain-controlled fatigue testing [Dowling]
Strain vs. life curves [Dowling]
6PSMN
ASME BPVC VIII Div 3: Design fatigue curves for non-welded machined parts made of forged carbon or low-alloy steel
MPa
7000
700
70
620 MPa
860-1200 MPa
Coffin-Manson’s equation
( )
( )( )
p
a f
e
a a f
a
Coffin-Manson's equation for the fatigue life for a fully reversed strain cycle:
2
Combining this with Basquin's equation,
2 ,
yields the Basquin-Coffin-Manson equation,
c c
b b
N
N CN
E E N BN
ε ε
ε σ σ
ε
′= =
′= = =
= ( )( ) ( )
( ) ( ) ( )
e p
a a f f
at t
a at t t
2 2 .
This may be rewritten in terms of the 'transition' parameters and :
2 .
b c b c
b c
E N N BN CN
N
N N N N
ε ε σ ε
ε
ε ε
′ ′+ = + = +
= +
7PSMN
Transition fatigue lives vs. hardnessfor a wide range of steels [Dowling].
335 1015 1725
Fatigue Limit
8PSMN
Fatigue limits of ferrous metals in rotating bending proportional to the tensile strength as long as Rm < 1400 MPa [Dowling].
Fatigue strengths in rotating bending at N = 5·108 cycles for wrought Al alloys approximately proportional to the tensile strength as long as Rm < 325 MPa [Dowling].
9PSMN
FKM fatigue ratios (and mean stress sensitivities)
W
W
W m
W W
f R
f
σ
τ
σ
τ σ
=
=
Mean Stress Effect
10PSMN
S-N curves for smooth specimens of an Al alloy under axial loading at various mean stresses [Dowling].
Haigh diagrams for smooth specimens of an Al alloy under axial loading at various lives [Dowling].
11PSMN
Normalised Haigh diagrams for smooth specimens of an Al alloy under axial loading showing the Goodman (solid) and Hempel-Morrow (long-dashed) lines and the Gerber parabola (short-dashed)[Dowling].
Mean stress effect on fatigue strength
( )
( )
a W m m
2
a W m m
a W m m
Goodman
1
Gerber
1
Gerber, generalised
1
N
N
N
R
R
Rδ
σ σ σ
σ σ σ
σ σ σ
+ =
+ =
+ =
( )
( )
a W m f
a W f m W
1
m a a W
Morrow
1
Ditto in terms of MSS
Walker (SWT, 0.5)
N
N N
N
γ γ
σ σ σ σ
σ σ σ σ σ
γ
σ σ σ σ−
′+ =
′+ =
=
+ =
12PSMN
Equivalent fully reversed stress amplitude
( )( )( )( )
( ) ( )
aar W f
m f
a f m
1
ar m a a W
Morrow
21
2
Walker
b
N
b
N
N
N
γ γ
σσ σ σ
σ σ
σ σ σ
σ σ σ σ σ−
′= = = ⇒′−
′= −
= + =
Haigh diagram at fatigue limit forSGCI EN-GJS-400-18-LT
13PSMN
Straight-line Haigh diagram at fatigue limit
according to FKM Guideline and Hempel-Morrow
Mean stress effect on fatigue limit
{ }
W W
a m W
W f m
W
Equations formally identical: fatigue strengths
are merely replaced by fatigue limits .
Hempel-Morrow in terms of FKM mean stress
sensitivity and fatigue ratio:
[MPa] 1000
N
M
M aR b
σ
σ
σ σ
σ σ σ
σ σ
σ
+ =
′= = +
=W mf Rσ
14PSMN
FKM (fatigue ratios and) mean stress sensitivities
W
m[MPa] 1000M aR b
M f M
σ
τ τ σ
= +
=
Goodman, Morrow and Gerber lines in Haigh diagram
0
100
200
300
400
500
600
700
800
900
1000
-1000 -500 0 500 1000
σmax = Rm
σmax = Re
σmin = -Re
σmin = 0
σmax = 0
Goodman
Gerber, σm > 0
Gerber, σm < 0
Morrow
σa, MPa
σm, MPa
15PSMN
Haigh diagram based on Walker’s equation
0
100
200
300
400
500
600
700
800
900
1000
-1000 -500 0 500 1000
σmax = Rm
σmax = Re
σmin = -Re
σmin = 0
σmax = 0
Walker, γ = 0
Walker, γ = 1
Walker, γ = 0,5
σm, MPa
σa, MPa
Fitting Hempel-Morrow line and Walker curve to R = -1 and R = 0 according to FKM Guideline for wrought steel
16PSMN
Hempel-Morrow parameter for wrought steel according to FKM Guideline and Walker
parameter fitted to R = -1 and R = 0
Scatter
17PSMN
Scatter in S-N data from rotating-bending specimens of an Al alloy [Dowling, Grover].
Statistical distribution (histogram) of fatigue lives for 57 smooth specimens of 7075-T6 aluminium tested at Sa = 207 MPa (30 ksi) in rotating bending [Dowling, Sinclair].
18PSMN
Rotating-bending S-N curves for various probabilities of failure for smooth specimens of 7075-T6 aluminium [Dowling, Sinclair].
Scatter in fatigue testing
Fatigue life scatter Fatigue strength scatter
19PSMN
Weibull Distribution
Waloddi Weibull (1887-1979) receives the Great Gold Medal of the Royal Swedish Academy of Engineering Sciences
Professor Weibull's proudest moment came in 1978 when he received the Great Gold medal which was
personally presented to him by King Carl XVI Gustav of Sweden. Below is the photo with King Carl XVI
Gustav of Sweden, Waloddi Weibull, and in the middle Gunar Hambræus, then President of the Royal
Swedish Academy of Engineering. When Waloddi stood in front of the King he said: "Seventy-one years ago I
stood in front of Your Majesty's grandfather's grandfather (King Oscar II) and got my officer's commission."
The King then said: "That is fantastic!"
20PSMN
Cumulative distribution function
The cumulative distribution functionof the three-parameter Weibull distribution
The cumulative distribution functionof a two-parameter Weibull distribution
λ = scale parameterk = shape parameterθ = location parameter
Two-parameter Weibull probability density function
0
2
4
6
8
10
12
0 0,5 1 1,5 2
k = 2 k = 5 k = 10 k = 20 k = 30
21PSMN
Mean and variance
Mean of the two-parameter Weibull distribution
Variance of the two-parameter Weibull distribution
Coefficient of variation, CVδ = σ/µ = √var(X)/E(X)
Weibull CV vs. Weibull exponent
22PSMN
Principle P-S-N diagram for smooth member under cyclic loading
( )( ) ( ) ( )
( )( )
( )( )( )
50
a
f a
1
a f m 50
50 a A 501
A 50 f m
Probability that failure for a stress amplitude of occurs within
cycles:
Pr 1 2
Basquin's equation:
2
2
Probability that failure at
nn N
N
m
m
Nm
N
n
P N n F n
Nn N
n
β
σ
σ
σ σ σσ σ
σ σ σ
−
−
−
= < = = −
′= − ⇒ =
′= −
( )( ) ( ) ( )a A 50 a A 50
a
f A a
a life of cycles occurs for a stress amplitude
below :
Pr 1 2 1 2m n
N N
N n
n
P n mβ βσσ σ σ σ
σ
σ
σ σ β β− −= < = − = − ⇒ =
P-S-N diagram based on Weibull life distribution
23PSMN
P-S-N diagram based on Weibull life distribution (2)
( )( )
( )( )
a A 50 m
f
1
1
a f m
f
Solving the equation for the probability of failure
1 2
for the stress amplitude yields
12 ln ln 2
1
N
m
P
NP
βσ
σ
σ σ σ
β
σ σ σ
−
−
= −
′= − −
P-S-N diagram for smooth member (ββββσσσσ = 30)under fully reversed loading
100
1000
1,00E+03 1,00E+04 1,00E+05 1,00E+06 1,00E+07 1,00E+08
Pf = 99%
Pf = 50%
Pf = 1%
Cycles to failure N
Stress amplitude σ
a, MPa
24PSMN
Safety factor based on Weibull distributed fatigue strength
( )( )
( )
a A 50 m
f
1
A 50
a f
Solving the equation for the Weibull probability of failure
1 2
for the safety factor
ln 2
ln 1 1
N
N
S
P
fP
βσ
σ
σ σ σ
β
σσ
−= −
= =
−
Safety factors based on Gaussian fatigue strength
( )
( )
a A 50
f A a
A 50
1
f
Similarly the equation for the Gaussian probability of failure
1 1Pr
1
1
coefficient of variation (= standard deviation/mean)
N S
N
N
S
fP
fP
σ σσ σ
δ σ δ
δ
δ
−
− − = < = Φ = Φ ⇒ ⋅
=+ ⋅Φ
=
25PSMN
Safety factors based on Weibull or Gauss distributed fatigue strength for coefficients of variation δδδδ = 0.025…0.1
References
� N.E. Dowling, Mechanical Behavior of Materials, 4th edition,
Prentice Hall, 2012.
� Rechnerischer Festigkeitsnachweis für Maschinenbauteile aus
Stahl, Eisenguss- und Aluminiumwerkstoffen. 6. Auflage, FKM,
Frankfurt, 2012. English version: Analytical strength assessment.