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.