gh9 multiaxial fatigueutmis.org.loopiadns.com/.../06/gh9_multiaxial_fatigue.pdf · 2018-11-02 ·...
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1Multiaxial
Lecture 9 Fatigue limit for
multiaxial stress cycles
Stress histories of marine diesel engine crankshaft
(courtesy Wärtsilä)
2Multiaxial
General multiaxial stress histories
General multiaxial stress histories
( ) ( )
T T
T2 T T 2
x xy xz
xy y yz
xz yz z
σ τ ττ σ ττ τ σ
σσ
τ σ σ σ
=
=
= −
= = − − = −
S
s = Sn
n s = n Sn
τ s n
τ τ s n s n s s
3Multiaxial
Henri Matisse (1869-1954): La Vague (1952)
Musée Matisse, Nice
Out-of-phase plane-stress cycles
( )( ) ( )( ) ( )
m a
m a
m a
sin ,
sin ,
sin ,
0.
x x x
y y y y
xy xy xy xy
xz yz z
t t
t t
t t
σ σ σ ω
σ σ σ ω α
τ τ τ ω α
τ τ σ
= +
= + −
= + −
= = =
4Multiaxial
Normal and shear stresses on φφφφ plane
( )( ) ( ) ( )
2 2
2 2
cos sin 2 sin cos ,
sin cos cos sin .
x y xy
y x xy
σ φ σ φ σ φ τ φ φ
τ φ σ σ φ φ τ φ φ
= + +
= − + −
Straight-line Haigh diagram at fatigue limit
according to FKM Guideline and Hempel-
Morrow
5Multiaxial
Haigh diagram according to FKM Guideline
W
W
W m
W W
f R
f
σ
τ
σ
τ σ
=
=
a W m
a W m
M
M
σ
τ
σ σ σ
τ τ τ
= −
= −
( )( )
W
W A A m
W A A
[MPa] 1000M aR b
M f M
σ
τ τ σ
σ σ σ
τ τ τ
= − = +
= − =
Empirical material parameters according to
FKM Guideline
6Multiaxial
Sines’ criterion at fatigue limit for proportional cycling
ar a, Mises 1m W
2 2 2
a, Mises a a a a a
1m m m
3x y x y xy
x y
M I
I
σσ σ σ
σ σ σ σ σ τ
σ σ
= + ≤
= + − +
= +
Critical-plane criteria at fatigue limit for
non-proportional cycling
( ) ( ){ }( ) ( ){ }( ){ }
ar a m W0 π
a max crit0 π
ar a 1m W0 π
Normal stress max
Findley max
Tresca-Sines max 2
M
f k f
M I
σφ
φ
σφ
σ σ φ σ φ σ
τ φ σ φ
σ τ φ σ
≤ ≤
≤ ≤
≤ ≤
= + ≤
= + ≤
= + ≤
( ) ( ){ } ( ){ }( ) ( ){ }
12a
00
max0
max ; min ;
max ;
t Tt T
t T
t t
t
τ φ τ φ τ φ
σ φ σ φ
≤ ≤≤ ≤
≤ ≤
= −
=
7Multiaxial
Findley’s criterion in terms of
conventional fatigue limits
( )( )( )( )
2
W
2
A
crit
2
W
2
A
2 1
2 2 1 4
1
1 4
k k
k kf
k
k
σ
σ
τ
τ
+ +
+ += + +
( ) ( )
( )a max
ar W0 π 21
2
max1
k
k kφ
τ φ σ φσ σ
≤ ≤
+
= ≤ + +
Findley’s fcrit in terms of ττττW
( )
( )
W
a a W
max max W
a max W W
max
W W
sin
Mohr's circle:
cos2 cos 2
sin 2 sin 2
Findley parameter:
cos 2 sin 2
Critical plane associated with is given by
2 sin 2 2 cos2 0 tan 2
cos2
xy
xy
xy
t
f k k
f
f k k
τ τ ω
τ τ φ τ φ
σ τ φ τ φ
φ τ σ τ φ τ φ
φ τ φ τ φ φ
=
= =
= =
= + = +
′ = − + = ⇒ =
2 2
2 2
max W W crit
2
crit W
1 1 , sin 2 1 1
max 1 1
1
k k
f f k k k k f
f k
φ
φ φ
τ τ
τ
∀
= + = +
= = + + ⋅ + = ⇒
= +
.
8Multiaxial
Fatigue parameters for Findley model
Fatigue parameters for wrought steel
9Multiaxial
ASME BPV-III-1 SSC
(simultaneous stress components) criterion
( ){ }( ) ( ) ( )
ar a Wˆ ˆ0 π, 0 ,
12a
ˆmax 2 , ;
ˆ ˆ, ; ; ;
t T t t Tt t
t t t t
φσ τ φ σ
τ φ τ φ τ φ
≤ ≤ ≤ ≤ ≤ ≤= ≤
= −
SSC formulation of the Findley criterion
( ) ( ){ }( ) ( ) ( )( ) ( ) ( ){ }( ) ( )
a max critˆ ˆ0 π, 0 ,
12a
max
a max
ˆ ˆmax , ; , ;
ˆ ˆ, ; ; ;
ˆ ˆ, ; max ; , ;
and now refer to the same instants in time,
making their (physical) interaction more likely.
t T t t Tf t t k t t f
t t t t
t t t t
φτ φ σ φ
τ φ τ φ τ φ
σ φ σ φ σ φ
τ φ σ φ
≤ ≤ ≤ ≤ ≤ ≤= + ≤
= −
=
10Multiaxial
Multiaxial fatigue criteria stated in
engineering design codes and standards
Criterion Code
Normal stress API RP 17G (ISO)
Findley -
Mises BPVC-VIII-2
Mises-Sines -
Tresca BPVC-III-1
Tresca-Sines BPVC-VIII-3*
*Mean stress correction based on mean normal stress on critical plane instead of I1m
Fatigue limits from n = 220 tension-torsion test
series compiled from nine different sources
11Multiaxial
Fatigue limit predictions for
symmetric tension-torsion cycles
( )( ) ( )( ) ( )
2
a W a W
22
a W a W
22
a W a W
Normal stress 1
Mises 3 1
Tresca 4 1
x xy
x xy
x xy
σ σ τ σ
σ σ τ σ
σ σ τ σ
+ =
+ =
+ =
Fatigue limit test data and predictions
for tension-torsion cycles
12Multiaxial
Fatigue limit predictions for
symmetric tension-torsion cycles
( ) ( )
ar W
1
2
1
,
1 ,
n
p ii
n
p i pi
p
m p n
s p m n
σ σ
=
=
=
=
= − −
∑
∑
Example: p = 0.8 unsafely predicts that σar has only reached 80% of the
value required for fatigue failure, although test data indicate that the
tension-torsion cycle is already at the fatigue limit.
Normalised predictions of σar at the fatigue limit
from different multiaxial fatigue criteria
for 220 tension-torsion cycles
13Multiaxial
Reference
� Ø. A. Bruun, Fatigue assessment of components subjected to non-proportional stress histories. MSc Thesis, NTNU, 2013. (Received the Prize of the Swedish Fatigue Network for the best Final Year Project in 2008.)
� Ø. A. Bruun, G. Härkegård, A comparative study of design code criteria for prediction of the fatigue limit under in-phase and out-of-phase tension-torsion cycles. Submitted to the International Journal of Fatigue.