multiaxial fatigue — a probabilistic analysis of …anek/research/oh_fatigue99.pdf3 (20)...

1 (20)Multiaxial fatigue Ð Probabilistic analysis

Anders Ekberg Solid Mechanics and CHARMEC, [email protected]/charmec/

Reine Lindqvist Naval Architecture, [email protected]

Martin Olofsson Mathematical Statistics, [email protected]/Centres/SC

Multiaxial Fatigue Ð a Probabilistic Analysis of Initiation in Cases of

Defined Stress Cycles


Statistical analysis of multi-axial fatiguePrinciple

Loads and material parameters are treated as stochastic variables

The Dang Van criterion (or similar) is used to analyze resulting (stochastic) equivalent stresses

AimDerive the probability of fatigue initiation

Solution methodsAnalytical solutions- Only possible for very simple cases

Numerical methods- Monte Carlo simulation combined with a

neural network modeling


Fatigue initiation in a pressure vesselModel

Stresses are derived using analytical expressions

Maximum pressure, ˜maxp , during a pressure cycle and material parameters c̃ and σ̃e are treated as stochastic variables (indicated by ̃)

Minimum pressure, pmin, during a pressure cycle is considered as deterministic and equal for all pressure cycles

Equivalent stresses are derived using the Dang Van criterion

Equivalent stresses are compared to a stochastic fatigue limit

FeaturesIn-phase loading

Fixed principle directions


Equivalent stresses in proportional loading

Stress components defined by internal pressure and geometry as

σ σθ1 = = ⋅p rt

, σ σ2 2= = ⋅

xp r

t and σ σ3 0= =z

In-phase, multi-axial loadingwith fixed principal directions ⇒ the Dang Van criterion can be expressed as

σσ σ

σ σDVa a

h,max e=−

+ ⋅ <1 3

2, , c

where c and σ e are materialparameters

t

2r x

z

u


Stochastic equivalent stressesIntroduce stochastic max-pressure and material parameters. The stochastic equivalent stress can then be expressed as

˜ ˜ ˜max, minσDV,i irt

c p p= +( ) −[ ]4

1 2

Consider a design life of N pressure cycles. Maximum equivalent stress and pressure during this life are denoted

˜max

( ˜ ),Si N i=

≤σDV and ˜

max( ˜ ),P

i Np ax i=

≤ m

and

˜ ˜ ˜minS

rt

c P p= +( ) −[ ]4

1 2

Given a distribution of ˜maxp and c̃ , what is the distribution of S̃?


Probability of fatigue initiationAssuming ˜max,p i independent for different igives

F x P x F xP pN

˜ ˜[ ˜ ]max

( ) = ≤ = ( )( )Prob and

f xx

F x N F x f xP P P

Np˜ ˜ ˜ ˜max

( ) = ( ) = ( )( ) ( )−dd

1

After a number of manipulations, (and some assumptions of variable independencies) the PDF of the largest equivalent stress during N pressure cycles is obtained as

fN

fu

Ftx

rup

uf

txru

pu

tru

uS c

u

u

p

N

p˜ ˜ ˜min

˜min

max max= −

⋅ +

⋅ +

=

=∞ −

∫21

24 4 4

1

1

d

and the probability of fatigue initiation is given as

Prob d de e˜ ˜ ˜ ˜S f u f v v u

S

u

>[ ] = ( ) ⋅ ( )∞

∫ ∫σ σ0 0


Probability density function

0 2 4 6 8 10 12 14x 107

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5 x 10-7

Equivalent stress magnitudeF

req

uen

cy

c̃ = [ ]N 0.125,0.02

˜maxp = [ ]N 5,0.5 [MPa]

c̃ ≈ ⋅[ ]N 0.125,1 10-3

˜maxp = [ ]N 5,0.5 [MPa]

0 2 4 6 8 10 12 14x 107

0

0.2

0.4

0.6

0.8

1

1.2

1.4 x 10-7

Equivalent stress magnitude

Fre

qu

ency

N = 1

N = 102

N = 104N = 106

N = 108


Equivalent stress distribution

In the picture, a distribution of S̃ for normally distributed˜maxp and c̃ is compared to a normal distribution. The shape of the distribution is dependent on the variance of c̃ .

Employing the Crosslandcriterion will give the same result as the Dang Van criterion

The error in the picture is exaggerated10 times

3 4 5 6 7 8 9x 107

0

1

2

3

4

5

6

7 x 10-8

Magnitude of equivalent stress

Fre

qu

ency

Ð)Simulated equivalent) stress distribution

--)Curve fitted normal) distribution

Rolling contact fatigue of railway wheelsModel

Hertzian contact with stochastic load magnitudesPertinent stochastic equivalent stresses derived using the Dang Van criterionNeural network modeling to reduce computational demandsEquivalent stresses compared to stochastic fatigue limitProbability of fatigue initiation is derivedSuggestion to account for limited damageaccumulation

Complicating featuresMulti-axial loadingRotating principal stress directionsOut-of phase loading


Q̃

˜ ˜σ σij ⇒ DV

X̃


Stochastic railway loads (log-normal)

Vertical load: ˜ ˜Q Q= +0

1eΞ

Horizontal load: ˜ ˜X X= +0

2eΞ

Ξ̃1 and Ξ̃2 arenormally distributed

Correlation defined as

ρ = Ξ ,Ξ Ξ ΞCov Var Var˜ ˜ ˜ ˜1 2 1 2( ) ( ) ⋅ ( ) 40 60 80 100 120 140 160 180

010

2030

4050

600

0.002

0.004

0.006

0.008

0.01

QX

Pro

babi

lty

~~


Equivalent stressesIn a case of general loading, the Dang Van criterion can be expressed as

σ σ σ σ σDVa a

h e=∈

( ) − ( ) + ⋅ ( )

<max

t Tt t

c td d1 3

2

where σ1ad t( ) and σ3a

d t( ) are the largest and smallest principal values of the tensor

σ σ σijd

ijd

ijdt t,a mid( ) = ( ) − ,

σ ijd

,mid is the tensor that fulfills the optimization criterion

σ σ σijd

ijd

ijd

t Tt, ,min

maxmid mid=

∈( ) −

This optimization has to be performed for every material point studied and for every load combination


Monte Carlo simulationAim

Derive a distribution of the equivalent stress, σ̃DV provided a distribution of applied loads Q̃ and X̃Especially the distribution of the extreme magnitude of σ̃DV during N load cycles is of interest

MethodMonte Carlo simulation using importance sampling

ProblemSome 107 calculations of equivalent stresses have to be performed

SolutionUse of neural network (NN) simulation


Neural network modeling

Verticalload

Lateralload

Transfer function F1:1Bias 1:1

Weight 1:1:1 Weight 1:2:1

Output =

Equivalent stress

logsig linear

F11 11 11: : : := +∑w I bii

i

TrainingExample cases are used to optimize w and b

EffectivityTraining takes some minutesA trained network calculates equivalent stresses instantaneously


Trained neural network

020

4060

80100

50100

150200

250300

350140

160

180

200

220

240

260

280

300

320

Y [kN]

DEEP ~ σEQ = f(Q,Y)

Q [kN]

σ EQ

[M

Pa]

X indicates result from physical model

Surface indicates the neural network model

Loads NN-model

σDV

LoadsPhysicalmodel σDV


Statistical simulation results

~4 mm:s

200 210 220 230 240 250 260 270 280 290 3000

0.02

0.04

0.06

0.08

0.1

0.12

Stress [MPa]

Pro

bab

ility

den

sity

r = 0

r = 0

r = 0.9

r = 0.9Fatigue lim

it

Largest equiv stress S~

104th largestequiv stressσDV(k)~

Distribution of loads

Distribution of σDV

NN Ð simulation+

Monte Carlo simulation

+Importance sampling

Distribution ofmaxi∈ N

σDV


Statistical simulation results, contÕdPosition of fatigue initiation

Mainly vertical loading ⇒ Some 4 mm:s below surfaceLarge frictional forces ⇒ At the surface

Both these positions werestudied in the simulation

SimulationsSimulations of stochastic equivalent stresses were made for material points at the surface and below the surfaceFor the case studied, the largest probability of fatigue initiation was predicted below the surface

at the surface


Concluding remarksIt is possible to analyze the probability of fatigue initiation in cases of stochastic, multi-axial loadingAnalytical analysis is possible in simple casesFor more complicated load cases, Monte Carlo simulations are feasibleNeural network modeling can be applied in order to decrease computational effortsThe results implicates that in many cases (e.g., in railway applications) only the most extreme load cases combined with the weakest material will induce fatigue damageTo estimate probabilities of fatigue initiation in stochastic loading, a knowledge of extreme loads and material strength is essentialA simple approach to fatigue design allowing a certain number load cycles to exceed the fatigue limit is outlined

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