Fatigue from Random Loads

Fred Kihm

HBM nCode

+33 1 30 18 20 20

Fred.kihm@hbmncode.com

The Leading Solution for Durability Engineering

• nCode products help:

Test engineers maximize the value of measured data through rapid analysis and collaborative data sharing.

Design engineers make the right engineering decisions to deliver optimized product performance.

Operations and asset managers understand product usage to reduce costs and avoid unexpected failures.

2

nCode Product Range

• Complex analysis to report, simply done

• Graphical, interactive & powerful analysis

• World leading fatigue analysis capabilities

• Enables collaboration, manages data, and automates standardized analysis

• Search, query and reporting through secure web access.

• Data to information to decisions

• Fatigue analysis technology for FEA

• Process encapsulation

• Fast, configurable, and scalable

Data Processing System for Durability

Streamlining the CAE Durability

Process

Maximizing ROI on Test and

Durability

Product Lifecycle Performance

Information

↓

Understanding

↓

Decisions

DESIGN

TEST OPERATION

• Physical validation

• Smarter testing

• Computer simulation

• Better designs, faster

• Understand usage

• Efficient operations

Content

1/ HBM nCode products presentation

2/ Introduction to Fatigue Analysis

3/ Deriving a representative Random Spectrum (PSD)

4/ Fatigue Damage from a PSD

Fatigue Analysis

An Introduction to Fatigue

• The 5 Box Trick :

Geometry

Material

Properties

Loading

Environment

Fatigue

Analysis

Fatigue

Results

The 5 Box Trick

3 Main Approaches:

• Stress-Life (SN Analysis)

• Strain-Life (EN or Crack Initiation Analysis)

• Crack Growth (LEFM)

Alternating Stress

Crystal surface

Slip bands form

along planes of

maximum shear

giving rise to

surface extrusions

and intrusions

Strain

Str

ess Failure mode 1:

Failure when stress

exceeds tensile

strength in a single

pass. This is not

fatigue failure.

Two failure modes

Time

Str

ess

Apply cyclic

load at low

stress level

Failure mode 2:

Failure occurs after a period of time

even though stress is low. The

component seems to get ‘tired’, hence

the name FATIGUE.

Transition from Stage I - II

Not all stage I cracks develop to stage II.

Most have insufficient energy to cross

the grain barriers. But it only takes one!

Fast fracture

Beacharks due to

crack propagation

Key Factors Affecting Fatigue

• Stress or Strain range The bigger the range the faster the failure

Fatigue life reduces exponentially with range

Str

ess

Ra

ng

e

• Mean Stress

Tensile mean accelerates fatigue

Compressive mean retards it

Viewed conceptually, a tensile mean is acting

to force open the crack while the

compressive mean is trying to keep it closed

The SN Curve

• ‘Infinite life’ region doesn’t really exist

• Aluminium doesn’t have an infinite life

• Steel with variable amplitude loading

doesn’t either

• SN method uses linear stress as input but

fatigue is driven by the release of plastic

shear STRAIN energy.

• Therefore SN is only valid when stress is

linearly related to strain, I.e. below yield.

Plastic

region

Tra

nsitio

n

b1

b2

Elastic

region

Infinite

life

region??

?

En

du

ran

ce

Dealing with Mean (Residual) Stresses

Viewed conceptually, a tensile

mean is acting to force open the

crack while the compressive

mean is trying to keep it closed

• Tensile mean accelerates fatigue

• Compressive mean retards it

Goodman Correction

Gerber Correction

Mean stress corrections are based

on modifications to the SN curve.

The curve is lowered in the

presence of a tensile mean and

raised in the presence of a

compressive mean

( ) b

f

u

m f N

S S

- = 2 1

`

2 1 s

s

( ) b

f

u

m f

N S

S

- = 2 1

2

`

2 1

s s

S = Stress range

sm = Mean stress

Su = Ultimate tensile strength

Signal statistics – Rainflow cycle analysis

• Fatigue tests are conducted under constant amplitude sinusoidal loading

• Real loading is usually fairly random

• Rainflow Cycle Counting is a technique that allows us to break down the real loads into equivalent cycles ranges so we can do fatigue analysis.

Sig

nal

Pro

cess

ing

Signal statistics – Rainflow cycle analysis

time

100

300

200

400

500

Peak Valley Extraction

time

100

300

200

400

500

Reorder to start

from ABS max Imagine the signal is

filled with water

time

100

300

200

400

500

Range Mean No.

time

100

300

200

400

500

45

0

22

5

Drain water starting at

lowest valley, measure

total & mean depth

drained

450 225 1

time

100

300

200

400

500

15

0

50

Continue by draining next

lowest, etc.

50 150 1

100 300 2

Sig

nal

Pro

cess

ing

Signal statistics – Rainflow cycle analysis

Range Mean No.

450 225 1

225

Range

1

2 No

. 300

100 300 2

150

50 150 1

• Take each cycle and create a 3D

histogram showing range vs. mean

vs. number of cycles counted...

Sig

nal

Pro

cess

ing

Range

N

100 MPa

60000

Material Life Curve

LifedamagedAccumulate %5.060000

300==

300 Cycles

=i f

i

N

NDamage

Damage Counting with Miner

etc...

Time History Peak Valley

Extraction

Rainflow Cycle

Counting

N

100

60000

Damage Counting Damage Histogram

LIFE

Fatigue Analysis Route

Str

ess o

r S

train

Time

Str

ess o

r S

train

Time

s

Mean stress

Strain Range

Statistical Nature of Fatigue (1/2)

Load history

Geometry

Fatigue Properties

Fatigue Analysis

Fatigue Results

Optimize

Scatter in

material data

Variable

production

quality

Unknown customer loading

Resulting statistical distribution of life

Statistical Nature of Fatigue (2/2)

Fatigue requires a well defined

Cycle distribution

Normalized Rainflow histograms

Road Load – 3 mns

Road Load – 200 mns

Load Histories

variables 57%

Material Properties

24%

Surface finish 16%

Stress Concentrators 3%

Fatigue is highly

sensitive to loads

Probabilistic Sensitivity

Test Tayloring to derive a representative PSD

What Do We Want From A Durability Test?

• Durability test that’s suitable

for the item in question:

a component,

sub-assembly

or a whole vehicle

• Test must replicate the same failure

mechanisms as seen in the real world

• Test should be representative of the real

loading environment

• Test should be accelerated where possible to reduce project time scales

and costs

• Test specification can be used in FE based virtual test or real physical

test (qualification tests)

30-Apr-13

Page 22 HBM

– nCode Kurt

Munson

What Steps Are Involved?

1. Duty or Mission Profiling

Find out what’s expected of the vehicle / aircraft / component

How long should it last?

Determine the ordinary loads that it’s likely to see every day

Determine the extraordinary loads it might see and be expected to survive

2. Test Synthesis

Synthesise a test that exhibits the same damage as the Mission Profile

What is Mission Profiling?

• The damage on most Automotive structural

components is dominated by deterministic time

events

• Aerospace components are usually dominated

by continuous stochastic processes

Take off

Cruise

Air combat

Intercept

Cruise

Descent

Land

-10

0

10

20

30

55.8 56 56.2 56.4 Time (s)

Acce

lera

tio

n (

g)

-0.2

0

0.2

0.4

100 200 300 Time (sec)

Acce

lera

tio

n (

g)

Deterministic

Stochastic

24

Vibration Test Synthesis

Test Specification

• Test can be expressed in the form of a PSD or

Swept Sine signal

• Test has same damage content as Mission

Profile

• Test accounts for statistical uncertainties

• Test is accelerated to reduce test duration

• Test is verified to ensure no unrealistically high

loads are induced

Mission Profile

x 200 rpts

x 200 hrs

x 100 rpts

+

+

etc…

25

Approach - ‘GAM EG-13’, ‘NATO AECTP-200’, ‘MILSTD 810-G’

• Measured data is transformed into the

Fatigue Damage Domain

• Mission (or Duty) profile is summed in the

Fatigue Damage Domain using Minor’s

rule and statistical safety factors applied

• Target Mission damage is transformed back to a

PSD, this is the accelerated test spectrum

• PSD can be transformed to time signal or Sine

Sweep if necessary

Fatigue Damage

Domain

Time Signal

PSD

Sine Sweep

PSD

Time Signal

Sine Sweep

Measured Input Accelerated Test Output

Fatigue

Transform

Inverse

Transform

Review of Theory

• Fatigue Damage Spectrum

(FDS)

– Damage vs. Frequency plot

– FDS steadily builds through

exposure to vibration

• Shock Response Spectrum

(SRS)

– Peak shock vs. Frequency plot

– SRS sensitive to extreme

amplitude peaks

Flight load data

Qualification

Flight load data

Qualification

27

Step 1

Damage Transformation

Process Flow

Calculate

SRS

Calculate

FDS Event

FDS

Event

SRS

Time Signal

PSD

Sine Sweep

Step 3

Test Synthesis

and

Validation

Invert FDS

Test

PSD

Test Time

Step 2

Mission Profiling Sum FDS Mission

FDS

Envelope

SRS

Mission

SRS

Test Specification

Calculate

SRS

Test

SRS

Validate

SRS

FDS = Fatigue Damage Spectrum, represents fatigue damage content of input signal

SRS = Shock Response Spectrum, represents maximum amplitude of input signal

Vibration Fatigue

Getting response PSD’s using the Harmonic Response Transfer Function

s(f) L(f)=1g

PSD of Input Load L1

× =

f

1g

f f

G²/Hz MPa²/Hz Stress PSD at Element

FRF(f) = Complex transfer

tensor for unit Load Case L(f)

Unitary Spectrum

|FRF(f)|²

30

PSD Rainflow Counter

Number of upward

zero crossings, E[0] = 3

Number of peaks, E[P] = 6

= upward zero crossing

= peak x

time Str

ess (

MP

a)

1 second

Time History

x x

x

x x

x

Stress2

Hz

Frequency, Hz

Gk(f)

fk

( ) = ffGfm n

n nth Moment of area under PSD:

Number of upward

zero crossings,

Number of peaks, (Theory of SO Rice, 1954)

2

4

0

20

m

mPE

m

mE

=

=

RMS RMS = m0

31

The Cycle Range – The Narrow Band Approach

Narrow band PSD & time signal Probability of peaks given

by Rayleigh distribution

(Theory of JS Bendat, 1964)

Rainflow cycle Peak

Range

=

-

0

2

8

04

m

S

em

STPEN

32

Limitation of Narrow Band Approach

time

time

Wide band time signal • The narrow band solution assumes that all positive peaks are matched with corresponding troughs of similar magnitude. Hence the red signal is transformed to the green signal.

33

• Stress cycle range (rainflow) is weighted sum of values at 2, 4, and 6 RMS stress

The Cycle Range – Steinberg

(Theory of Steinberg)

N(S) = E[P]*T*

0mRMS =

0.683 at 2*RMS

0.271 at 4*RMS

0.043 at 6*RMS

34

The Cycle Range – Dirlik (1985)

( ) ( )4210 ,,, mmmmfSp D =

• Stress cycle range (rainflow) is weighted sum of Rayleigh and Exponential distributions developed from an extensive Monte Carlo simulation

where; zS

m=

2 0

=

m

m m

2

0 4

xm

m

m

mm = 1

0

2

4

( )D

xm

1

2

2

2

1=

-

DD D

R2

1 1

21

1=

- -

-

D D D3 1 21= - -( )

QD D R

D=

- - 1 25 3 2

1

. Rx D

D D

m=- -

- -

1

2

1 1

21

( )p S

D

Qe

D Z

Re D Z e

mD

Z

Q

Z

R

Z

=

- -

-

1 2

2

23

2

0

2

2

2

2

where; zS

m=

2 0

=

m

m m

2

0 4

xm

m

m

mm = 1

0

2

4

( )D

xm

1

2

2

2

1=

-

DD D

R2

1 1

21

1=

- -

-

D D D3 1 21= - -( )

QD D R

D=

- - 1 25 3 2

1

. Rx D

D D

m=- -

- -

1

2

1 1

21

( )p S

D

Qe

D Z

Re D Z e

mD

Z

Q

Z

R

Z

=

- -

-

1 2

2

23

2

0

2

2

2

2

35

• Stress cycle range (rainflow) is weighted sum of Rayleigh and Gaussian distributions

The Cycle Range – Lalanne/Rice (2000)

(Theory of Lalanne, 2000)

FE-Based Vibration Fatigue Analysis : summary

DesignLife PSD Vibration Results

Validation – notched vibration specimen

• Notched vibration specimen fixed to test rig at left hand end and vibrated vertically

• Fatigue life estimated by FE analysis and compared with test rig. Specimen Id Test Life (seconds)

1 1620

2 1020

3 1560

4 1560

5 1800

6 1860

7 2100

Average test life 1646

Rice/Lalanne 1980

Dirlik 1010

Narrow band 1080

Steinberg 5050

Conclusion Test Tayloring & Virtual Shaker Table

• Create accelerated vibration shaker tests from multiple measured data sets without exceeding realistic shock levels.

• Save time per test to reduce cost and increase throughput.

• Improves confidence in results by more closely representing the real loading environment.

• Uses a fatigue damage spectrum approach and has many advantages over traditional methods of enveloping PSDs.

• Simulate shaker tests up-front from FE results thus closing the loop for more efficient and right-first-time testing.

Thank you

HBM, Inc. (HBM-nCode)

Travelers Tower 1

26555 Evergreen Rd, Ste. 700

Southfield, MI 48076

Tel: (248) 350 8300

Toll free: (877) 737 4242

HBM United Kingdom Limited (HBM-nCode) Innovation

AMP Technology Centre

Brunel Way, Catcliffe

Rotherham

South Yorkshire

S60 5WG

Tel: +44 (0)114 275 5292

Fred.Kihm@hbmncode.com

01 30 18 20 20