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Fatigue from Random Loads
Fred Kihm
HBM nCode
+33 1 30 18 20 20
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
01 30 18 20 20