Fatigue Analysis using A/CSD Matrix – Feasibility Studyutmis.org.loopiadns.com/media/2016/08/Contribution... · Fatigue Analysis using A/CSD Matrix Proposed Load Model MDOF PSD

1 Fatigue Analysis using A/CSD Matrix

Fatigue Analysis using A/CSD

Matrix – Feasibility Study

AR16-016

Karl Holmgren

05-10-16


Presentation Scope

Current vs proposed analysis approach

Proposed methodology – general

Damage correlation

1DOF load

6DOF load

Weld variance vs load variance

Further work / questions


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


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


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?


General Methodology


General Methodology • Process

Mission

Profiling

Test

Synthesis

Response

Calc

Fatigue

Analysis


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


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


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


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


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


𝐶𝑆𝐷12

𝐶𝑆𝐷21

𝐶𝑆𝐷𝑚1

𝐶𝑆𝐷1𝑚 vehicle n

𝐴𝑆𝐷11

𝐴𝑆𝐷22


𝐶𝑆𝐷12

𝐶𝑆𝐷21

𝐶𝑆𝐷𝑚1

𝐶𝑆𝐷1𝑚

Probabilistic load

𝐴𝑆𝐷11

𝐴𝑆𝐷22


𝐶𝑆𝐷12

𝐶𝑆𝐷21

𝐶𝑆𝐷𝑚1

𝐶𝑆𝐷1𝑚

• Test synthesis, probabilistic load model


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.


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


Weld Fatigue Model Stress-Life curve PDF • Fatigue analysis


Damage Correlation

1DOF Load


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


Individual DOF • Individual Events

• Damage Equivalent PSD’s



Event axis

19 events



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


Individual DOF • Complete test




1.6

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0


Damage Correlation

6DOF Load


Correlated Load DOF • Individual Events


damage of the Time History Event axis

19 events



Correlated Load DOF • Complete test





Results


Roll

S/N curve position

damage

PDF


Roll Cumulative Probability Density Function

Load severity outcome

Weld fatigue model HAS variance

Weld fatigue model HAS NO variance

damage


Roll

Deterministic fat model

Probabilistic fat model

CDF (Cumulative Distribution Function)

damage


6DOF

damage

S/N curve position


6DOF

damage

Load severity outcome


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


Further Work / Questions


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


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?

Documents

Fatigue Analysis using A/CSD Matrix – Feasibility Studyutmis.org.loopiadns.com/media/2016/08/Contribution... · Fatigue Analysis using A/CSD Matrix Proposed Load Model MDOF PSD