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1 Fatigue Analysis using A/CSD Matrix
Fatigue Analysis using A/CSD
Matrix – Feasibility Study
AR16-016
Karl Holmgren
05-10-16
2 Fatigue Analysis using A/CSD Matrix
Presentation Scope
Current vs proposed analysis approach
Proposed methodology – general
Damage correlation
1DOF load
6DOF load
Weld variance vs load variance
Further work / questions
3 Fatigue Analysis using A/CSD Matrix
TitanX Vibration Validation Process
Physical prototype
Physical vehicle
Destructive vib test
• 7-axis (6DOF + frame twist)
• Pressure and Temp conditioning
of test specimen
• Acceleration
• Strain
• Temperature
• Pressure
In-vehicle DAQ
FEA
• Static inertia
Coupled with rough
stress acceptance criteria
• Modal
4 Fatigue Analysis using A/CSD Matrix
Method Deficiencies
The current FEA load types cannot consider:
- Cross correlation between load DOF
- Dynamic behavior of cooling module
Desired Properties of Load Model
Consider cross correlation between load DOF
Support a dynamic FEA
Be formulated in a probabilistic manner, to describe a specific vehicle type/application.
Examples: US vocational, or European distribution
5 Fatigue Analysis using A/CSD Matrix
Proposed Load Model
MDOF PSD
- PSDs and CSDs described using stochastic models
- This would enable us to target a specific level of confidence.
Challenges / Questions
The load time histories are non-stationary and largely non-random.
The PSD based fatigue methods assume the opposite.
Apart from the loads themselves, the fatigue models of welds and other materials
are best defined as probabilistic. For an MDOF load, this may necessitate a Monte-Carlo
facilitated fatigue analysis, to produce a joint probability distribution.
Despite this, the method must produce damage equivalent responses.
Is this necessary, or will the variability of either the load or the fatigue model dominate?
6 Fatigue Analysis using A/CSD Matrix
General Methodology
7 Fatigue Analysis using A/CSD Matrix
General Methodology • Process
Mission
Profiling
Test
Synthesis
Response
Calc
Fatigue
Analysis
8 Fatigue Analysis using A/CSD Matrix
General Methodology • Mission Profiling
Chassis responses of 7 vocational vehicles of one individual OEM used.
The same duty schedule for all 7 vehicles
The duty schedule is composed of a number of test track events, for which the OEM constructs
a balanced repetition map for each individual vehicle configuration tested.
Real world event 1
Real world event 2
Real world event m
Repeats 1
Repeats 2
Repeats, m
Gauge1 Gauge 2 Gauge n
CD11 CD12 CD1n
CD21 CD22 CD2n
CDm1 CDm2 CDmn
Test track event 1
Test track event 2
Test track event o
Repeats 1
Repeats 2
Repeats, m
Gauge1 Gauge 2 Gauge n
CD11 CD12 CD1n
CD21 CD22 CD2n
CDo1 CDm2 CDon
Damage and maximum response
equivalence
Calibration
parameters
9 Fatigue Analysis using A/CSD Matrix
General Methodology • Mission Profiling, Vehicle Configurations
Drive configurations: 8x6, 6x4, 6x2
6 trucks (rigids), 1 tractor
Front suspension ratings from 6 to 10 tonne
Rear suspension ratings from 21 to 35 tonne
Wheel bases from 4780 to 6810 mm
Frame rail thickness 6-11mm, height 250-300mm
Engine displacement from 11l to 15l
10 Fatigue Analysis using A/CSD Matrix
General Methodology • Test synthesis, general
Two approaches are attempted
Approach1: Standard averaged PSDs for diagonal terms
Approach2: Damage equivalent PSDs for diagonal terms
Standard, averaged PSDs used for cross terms in both approaches
Only FDS are considered – no specific consideration to maximum responses.
Time Signal
n events
FDS calculation
Q=10
displacement resp
b=-0.2
FDS
& event time
n events
Inverse of
freq domain
FDS calc
Dam equiv PSD
& test time
Time Signal
n events
Standard,
averaged
PSD calc
PSD
& test time
Approach 2
Approach 1
11 Fatigue Analysis using A/CSD Matrix
General Methodology
The level of the spectra require a certain test time to produce damage equivalence with
the original time histories.
If this test time varies between different vehicle signals, a test time normalization is
required before a probabilistic load may be defined.
We will normalize by multiplying the amplitude of the PSD by a coefficient, k.
𝐶𝐷 𝑡𝑒𝑠𝑡𝑖 = 𝐶𝐷 𝑡𝑒𝑠𝑡𝑖,𝑛𝑜𝑟𝑚
𝑇𝑖 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑖 = 𝑇𝑖,𝑛𝑜𝑟𝑚 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑖,𝑛𝑜𝑟𝑚
𝐶𝐷 𝑃𝑆𝐷𝑖,𝑛𝑜𝑟𝑚 =𝑇𝑖𝑇𝑖,𝑛𝑜𝑟𝑚
𝐶𝐷 𝑃𝑆𝐷𝑖
𝐶𝐷 𝑃𝑆𝐷𝑖 ~𝑅𝑀𝑆𝑖−1/𝑏
𝐶𝐷 𝑃𝑆𝐷𝑖,𝑛𝑜𝑟𝑚 = 𝐶𝐷 𝑘 ∙ 𝑃𝑆𝐷𝑖 ~𝑘−1/2𝑏 ∙ 𝑅𝑀𝑆𝑖
−1/𝑏
𝑘 =𝑇𝑖𝑇𝑖,𝑛𝑜𝑟𝑚
−2𝑏
• Test synthesis, normalization using test length
12 Fatigue Analysis using A/CSD Matrix
General Methodology
Based on n vehicle spectrum matrices, calculate a single spectrum matrix, where:
- Amplitude is approximated using a log-normal distribution
Where:
is the average of the natural logarithm of the amplitude for the n vehicles.
is the standard deviation of the natural logarithm of the amplitude for the n vehicles.
- Phase is approximated as a gaussian distribution:
Where:
is the average of the amplitude for the n vehicles.
is the standard deviation of the amplitude for the
n vehicles.
𝑎 𝑓 = 𝑙𝑛𝑁 𝜇 𝑓 , 𝜎 𝑓 2
𝜇 𝜎
𝜑 𝑓 = 𝑁 𝜇 𝑓 , 𝜎 𝑓 2
𝜇 𝜎
vehicle 1
𝐴𝑆𝐷11
𝐴𝑆𝐷22
𝐴𝑆𝐷𝑚𝑚
𝐶𝑆𝐷12
𝐶𝑆𝐷21
𝐶𝑆𝐷𝑚1
𝐶𝑆𝐷1𝑚 vehicle 2
𝐴𝑆𝐷11
𝐴𝑆𝐷22
𝐴𝑆𝐷𝑚𝑚
𝐶𝑆𝐷12
𝐶𝑆𝐷21
𝐶𝑆𝐷𝑚1
𝐶𝑆𝐷1𝑚 vehicle n
𝐴𝑆𝐷11
𝐴𝑆𝐷22
𝐴𝑆𝐷𝑚𝑚
𝐶𝑆𝐷12
𝐶𝑆𝐷21
𝐶𝑆𝐷𝑚1
𝐶𝑆𝐷1𝑚
Probabilistic load
𝐴𝑆𝐷11
𝐴𝑆𝐷22
𝐴𝑆𝐷𝑚𝑚
𝐶𝑆𝐷12
𝐶𝑆𝐷21
𝐶𝑆𝐷𝑚1
𝐶𝑆𝐷1𝑚
• Test synthesis, probabilistic load model
13 Fatigue Analysis using A/CSD Matrix
General Methodology • System Response, (static)
𝐺𝑧𝑧 𝑓 = 𝐻𝑎 𝑓 ∙ 𝐻𝑏 𝑓 ∙ 𝑊𝑎𝑏 𝑓
6
𝑏=1
6
𝑎=1
The frequency response is calculated as:
Where: is the FRF of load DOF a.
is the cross spectrum of load DOF a and b.
𝐻𝑎
𝑊𝑎𝑏
In order to emulate a static FEA we used an FRF with unit amplitude and zero phase for all load DOF.
The stress calculation, preceding an MDOF PSD facilitated fatigue analysis boils down to a
linear scale and combine of load PSD’s and CSD’s. Use of unit FRFs and zero phase implies
unit scaling coefficient vector.
In a real FEA-supported analysis the phase would still be zero and the scaling coefficient vector
would still be constant over the frequency range. However, the latter would vary with the response
point analyzed.
14 Fatigue Analysis using A/CSD Matrix
British Standard 7608:1993
Fatigue Design and Assessment of Steel Structures
For idealized hot-spot stress (class T)
𝑙𝑛 𝑁 = 𝑙𝑛 𝐶0 − 𝑑 ∙ 𝜎 − 𝑚 ∙ 𝑙𝑛 𝑆𝑟
number of cycles
weld joint constant
stDev of ln(N)
number of stDevs
deviating from 50%
in the negative direction
S-N slope (-1/b)
stress range
𝑁 = 𝑒𝑥𝑝 𝑙𝑛𝐶0𝑆𝑟𝑚 − 𝑑 ∙ 𝜎
General Methodology
𝑍 = 𝑒𝑥𝑝 𝜇 + 𝑋 ∙ 𝜎
Base expression for
log-normal distribution
𝑙𝑛 𝐶0 = 12.66
𝑚 = 3.0
𝜎 = 0.572
𝜇 = 𝑙𝑛𝐶0𝑆𝑟𝑚
Hence:
𝑋 = 𝑑
• Fatigue analysis
15 Fatigue Analysis using A/CSD Matrix
Weld Fatigue Model Stress-Life curve PDF • Fatigue analysis
16 Fatigue Analysis using A/CSD Matrix
Damage Correlation
1DOF Load
17 Fatigue Analysis using A/CSD Matrix
Individual DOF • Individual Events
• Standard, Averaged PSD’s
Damage normalized to the
damage of the Time History
Event axis
19 events
Damage of the Time History
18 Fatigue Analysis using A/CSD Matrix
Individual DOF • Individual Events
• Damage Equivalent PSD’s
Damage normalized to the
damage of the Time History
Event axis
19 events
Damage of the Time History
19 Fatigue Analysis using A/CSD Matrix
Individual DOF • Complete test, 4 formulations
Sum of the damages incurred by the 19 individual events,
with the load described using standard PSD’s. 𝐶𝐷19𝑃𝑆𝐷 = 𝑇𝑖 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑖
19
𝑖=1
Sum of the damages incurred by the 19 individual events,
with the load described using damage equivalent PSD’s.
𝐶𝐷19𝑃𝑆𝐷𝑒𝑞 = 𝑇𝑖 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑒𝑞𝑖
19
𝑖=1
The damage incurred by a single averaged PSD,
calculated as the averaged PSD of the entire test.
𝐶𝐷1𝑃𝑆𝐷 = 𝑇𝑡𝑒𝑠𝑡 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑡𝑒𝑠𝑡
𝑃𝑆𝐷𝑡𝑒𝑠𝑡 =1
𝑇𝑡𝑒𝑠𝑡 𝑇𝑖 ∙ 𝑃𝑆𝐷𝑖
19
𝑖=1
The damage incurred by a single averaged PSD,
calculated as the averaged damage equivalent PSD
of the entire test.
𝐶𝐷1𝑃𝑆𝐷 = 𝑇𝑡𝑒𝑠𝑡 ∙ 𝐶𝐷 𝑃𝑆𝐷𝑒𝑞𝑡𝑒𝑠𝑡
𝑃𝑆𝐷𝑒𝑞𝑡𝑒𝑠𝑡 =1
𝑇𝑡𝑒𝑠𝑡 𝑇𝑖 ∙ 𝑃𝑆𝐷𝑒𝑞𝑖
19
𝑖=1
20 Fatigue Analysis using A/CSD Matrix
Individual DOF • Complete test
Damage normalized to the
damage of the Time History
Damage of the Time History
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0
21 Fatigue Analysis using A/CSD Matrix
Damage Correlation
6DOF Load
22 Fatigue Analysis using A/CSD Matrix
Correlated Load DOF • Individual Events
Damage normalized to the
damage of the Time History Event axis
19 events
Damage of the Time History
23 Fatigue Analysis using A/CSD Matrix
Correlated Load DOF • Complete test
Damage normalized to the
damage of the Time History
Damage of the Time History
24 Fatigue Analysis using A/CSD Matrix
Results
25 Fatigue Analysis using A/CSD Matrix
Roll
S/N curve position
damage
26 Fatigue Analysis using A/CSD Matrix
Roll Cumulative Probability Density Function
Load severity outcome
Weld fatigue model HAS variance
Weld fatigue model HAS NO variance
damage
27 Fatigue Analysis using A/CSD Matrix
Roll
Deterministic fat model
Probabilistic fat model
CDF (Cumulative Distribution Function)
damage
28 Fatigue Analysis using A/CSD Matrix
6DOF
damage
S/N curve position
29 Fatigue Analysis using A/CSD Matrix
6DOF
damage
Load severity outcome
30 Fatigue Analysis using A/CSD Matrix
6DOF
50% failures
10% failures
1% failures
2000x damage
12.2x stress range
CDF (Cumulative Distribution Function)
PDF (Probability Distribution Function)
rel. damage
160x damage
5.3x stress range
Deterministic fat model
Probabilistic fat model
31 Fatigue Analysis using A/CSD Matrix
Further Work / Questions
32 Fatigue Analysis using A/CSD Matrix
Conclusions
For quasi static response analyses, MDOF PSD analysis with:
- damage equivalent PSDs used as diagonal terms.
- standard, averaged cross spectra used as cross terms.
provide a reasonable approximation of the original MDOF load time history.
The load variance overwhelmingly controls the variance of the response damage.
No necessity formulating the weld model as a probabilistic variable in a MCA*.
Only load model left as probabilistic variable in the MCA.
For fully correlated load severity outcomes for MDOF loads, no MCA needed.
For the collection of vehicle signals used, the resulting response damage variance
is greater than desired, risking either too expensive designs – or high risk of failure.
*MCA: Monte Carlo Analysis
33 Fatigue Analysis using A/CSD Matrix
Further Work / Questions
The load variance resulting from the 7 vocational vehicles used is great enough
to question the use of the model.
We have shown reasonable correlation between time domain response and
frequency domain response of quasi-static systems.
For Monte Carlo simulations with 6 loading DOF we have assumed the severity
outcome of all 6 loading DOF is fully correlated.
How do we construct load models of more reasonable variance?
Different vehicle type? Different type of test schedule design methodology?
Will the response of dynamic systems show as good correlation?
What happens to the fatigue life variance when using a more reasonable
correlation model?