06
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Institut für Maschinenelemente, Universität Stuttgart
Technologie Transfer Initiative, Universität Stuttgart
M. Sc. Martin Dazer
Dipl.-Ing. Stefan Kemmler
Prof. Dr.-Ing. Bernd Bertsche
Institute of Machine Components
Reliability Engineering
Technology Transfer Initiative
Dr.-Ing. Tobias Leopold
Dipl.-Ing Jens Fricke
Knorr-Bremse Group
Systeme für Nutzfahrzeuge GmbH
Rob
ust
Se
nsitiv
e
Virtual
Lifetime
Determination
Durability (t-t0) 106 LC
Fa
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(t)
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TechnologieTransferInitiative
STOCHASTIC SIMULATION APPROACH FOR
REALISTIC LIFETIME FORECAST AND ASSURANCE
OF COMMERCIAL VEHICLE BRAKING SYSTEMS
12. Weimar Optimization and Stochastic Days 2015
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Research Cooperation
2
Rob
ust
Se
nsitiv
e
Virtual
Lifetime
Determination
Durability (t-t0) 106 LC
Fa
ilu
re P
rob
ab
ilit
y F
(t)
Sh
ap
e p
ara
me
ter
b
TechnologieTransferInitiative
06
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Institut für Maschinenelemente, Universität Stuttgart
Technologie Transfer Initiative, Universität Stuttgart
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Agenda
1. Motivation
2. Application example: brake caliper
3. Simulation process
4. Design of Experiments
5. Result evaluation
6. Summary and Outlook
3
06
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Institut für Maschinenelemente, Universität Stuttgart
Technologie Transfer Initiative, Universität Stuttgart
1. MOTIVATION
Virtual Lifetime Determination of
commercial vehicle braking systems
4
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Identification of specific product
properties:
Failure mechanism
Lowest and longest cycle times
Failure distribution
Control of correct target-
engineering by requirements
1. Motivation
5
Real durability tests
Systematic and
unsystematic
scattering of
lifetime
tmin
tmax
Scattering produces a
characteristic lifetime distribution
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1. Motivation
6
Load cycles N (log)Str
ess A
mp
litu
de
(lo
g)
Occurring nominal load
amplitude on component
50%Endurable nominal load
amplitude in operation
Nominal load cycles Nnom (log)
cumulativedensity
Str
ess A
mp
litu
de
(lo
g)
10%
90%
10%
50%
90%
Scatter band of the load
amplitudes occur on the
component
Scatter band of endurable load
amplitudes in operation
Motivation:
Realistic lifetime prediction through
stochastic simulation of stress and
strength
Load cycles N (log)
F(t
) Scatter band of load cycles N
Stochastic
Simulation
Source: Bertsche 2004
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1. Motivation
7
Input
Variability
Output
VariabilitySimulation
System
Parameter variations of the product lead to variations in product properties as a
result of internal correlations.
This can cause functional or structural failure and the deterioration of product
quality.
Aim: Realistic forecast of time to failure
Lifetime
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1. Motivation
8
Parametric CAD-Model
Mapping the caliper geometry
Changes in the geometry
FEA-Simulation model
Mapping of damage-related effects
Stress amplitudes
Strain amplitudes
Parameter study
Automatic simulation of
the established DOE
Statistical evaluation
Source: Dynardo 2015
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Institut für Maschinenelemente, Universität Stuttgart
Technologie Transfer Initiative, Universität Stuttgart
2. APPLICATION EXAMPLE
BRAKE CALIPER
Virtual Lifetime Determination of
commercial vehicle braking systems
9
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10
Brake caliper
Application example:
Brake caliper
Pressure
cylinder dummy
Brake discBrake pads
2. Application example Brake caliper
Piston
Piston
Have to withstand high loads in
case of overload
Therefore tested with high loads
Failures in Low-Cycle-Fatigue
range (LCF)
0,9
0,8
0,7
0,6
0,5
0,4
0,3
0,2
0,150,
10,
09
99
90
8070605040
30
20
10
5
3
2
1
Standardized lifetime
Pe
rce
nt
Shape 2,89116
Scale 0,465464
A v erage 0,414998
Stdv . 0,155908
Median 0,410044
Statistics
Probability plot for lifetime
Full data - ML-Estimation
Weibull
Probability function of lifetime
Weibull
Full data – ML-Estimation
Standardized lifetime
Pe
rce
nt
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Institut für Maschinenelemente, Universität Stuttgart
Technologie Transfer Initiative, Universität Stuttgart
3. SIMULATION PROCESS
Virtual Lifetime Determination of
commercial vehicle braking systems
11
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3. Simulation process
12
Loadcase simulation
Load
definition
Lifetime distribution
Carrying
Capacity
Material
properties
Werkstoff-
charakterisierung
Bauteil-
geometrie
Beanspruchungs-
Zeitfunktion
Zyklische Spannungs
Dehnungskurve
Dehnungswöhlerlinie
FEM-Analyse Schädigungsparameter
Schädigungsparameter-
wöhlerlinie
SchädigungsrechnungKlassierung des
Lastkollekivs
Bauteillebensdauer
N
εA
elastisch
plastisch
Gesamt
PSWTPB
σA, σm,
ε A
F N
N
εA
120000116000112000108000
12
10
8
6
4
2
0
Belastung
Hä
ufi
gke
it
Belastungsverteilung
La
sta
mp
litu
de
n1 n2 n3N
n1 n2 n3
La
sta
mp
litu
de
Stress / Strain
hysteresis
Definition of claim and the associated
amplitudes Characterization of the relevant
material properties
Deterioration
Stochastic Simulation for variance determination
σa
εa
P =
10 %
50 %
90 %
εa
N
Pü =
10 %50 %
90 %
10 %
50 %
90 %
Pü =
Dehnungswöhlerlinie
The level of
deterioration related
stress and strain
amplitudes
Stress / Strain Wöhler
curve
Identification and
consideration of non-
homogeneous loading
states
Strength
Tension Pressure Shear Bending Torsion
Characterization of the
strength model
Source: maschinenbau-wissen 2015Source: Haibach 2005
Source: SOFEA 2015
Source: Haibach 2005
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13
3. Simulation process – load definition
Real force flow in the
brake caliper
Simulated force flow in
the brake caliper
Bolt pretension for lower
FE-calculation time
Contact surface
Contact surface
Clamping
force
system boundary
Clamping force causes
high FE-calculation time
system boundary
Same damage
state
System
simulation to
determine
deterioration
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3. Simulation process – load definition
14
There are variations in the clamping force occuring through
hysteresis effects
Load spectrum is determined from test reports
Best-Fit: Lognormal distribution
1,151,11,0510,95
99
95
80
50
20
5
1
Standardized clamping force
Pe
rce
nt
Test for goodness of fit
Lognormal
A D = 0,242
p-v alue = 0,759
Probability function for clamping force
Lognormal - 95%-KI
1,111,081,051,020,990,96
10
8
6
4
2
0
Standardized clamping force
De
nsit
iy
Shape 0,03424
Scale 0,03219
N 50
Lognormal
Densitiy function of clamping force
Standardized clamping force
Distribution of clamping force
LognormalProbability function of clamping force
Lognormal – 95 % KI
Standardized clamping force
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3. Simulation process
15
Loadcase simulation
Load
definition
Lifetime distribution
Carrying
capacity
Material
properties
Werkstoff-
charakterisierung
Bauteil-
geometrie
Beanspruchungs-
Zeitfunktion
Zyklische Spannungs
Dehnungskurve
Dehnungswöhlerlinie
FEM-Analyse Schädigungsparameter
Schädigungsparameter-
wöhlerlinie
SchädigungsrechnungKlassierung des
Lastkollekivs
Bauteillebensdauer
N
εA
elastisch
plastisch
Gesamt
PSWTPB
σA, σm,
ε A
F N
N
εA
120000116000112000108000
12
10
8
6
4
2
0
Belastung
Hä
ufi
gke
it
Belastungsverteilung
La
sta
mp
litu
de
n1 n2 n3N
n1 n2 n3
La
sta
mp
litu
de
Stress / Strain
hysteresis
Definition of claim and the associated
amplitudes Characterization of the relevant
material properties
Deterioration
Stochastic Simulation for variance determination
σa
εa
P =
10 %
50 %
90 %
εa
N
Pü =
10 %50 %
90 %
10 %
50 %
90 %
Pü =
Dehnungswöhlerlinie
Stress / Strain Wöhler
curve
Identification and
consideration of non-
homogeneous loading
states
Strength
Tension Pressure Shear Bending Torsion
Characterization of the
strength model
The level of
deterioration related
stress and strain
amplitudes
Source: Haibach 2005
Source: Haibach 2005
Source: SOFEA 2015
Source: maschinenbau-wissen 2015
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Technologie Transfer Initiative, Universität Stuttgart
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0
100
200
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8 9 10
Str
ess σ
[Mpa
]
Strain ε [%]
Material behavior GJS-600-6
Cyclic First load First discharge
3. Simulation process – material properties
16
Re = 370 MPa Significantly higher cyclic
yield strength Re‘
Mathematical modeling using
Ramberg-Osgood-equation:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾
1𝑛
R‘e = 500 MPa
Material shows clear
solidifying behaviorRe = first load yield strength
Re‘ = cyclic yield strength
General Ramberg-Osgood-
equation:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾
1𝑛
Modelling the fist load curve:
𝐾 = 1160𝑛 = 0,19
Modelling the first discharge
curve:
𝐾′′ = 1300𝑛′′ = 0,17
Modelling the cyclic curve:
𝐾′ = 924,9𝑛′ = 0,1
Nominal material behavior
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3. Simulation process – material properties
17
Re = 370 MPa
Deutlich höhere zyklische
Streckgrenze Re‘
Mathematische Modellierung
mit Ramberg-Osgood-
Gleichung
R‘e = 500 MPa
Werkstoff zeigt deutliches
verfestigendes VerhaltenRe = zügige Streckgrenze
Re‘ = zyklische Streckgrenze
0
100
200
300
400
500
600
700
800
900
1000
0 1 2 3 4 5 6 7 8 9 10
Str
ess σ
[Mpa
]
Strain ε [%]
Material behavior GJS-600-6
First load
How can the scattering
behavior mapped
mathematically?
General Ramberg-Osgood-
equation:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾
1𝑛
Modelling the fist load curve:
𝐾 = 1160𝑛 = 0,19
Modelling the first discharge
curve:
𝐾′′ = 1300𝑛′′ = 0,17
Modelling the cyclic curve:
𝐾′ = 924,9𝑛′ = 0,1
Nominal material behavior
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σa
εaεg,min εg,max
Rm
Rm,min
Rm,max
ε0,2
Rp0,2
Rp0,2;min
Rp0,2;max
ε0,2,min ε0,2,max
3. Simulation process – material properties
Only data from the static tensile test available
Mathematical derivation of the scattering of n and k
18
Dis
trib
utio
n o
fsm
alle
ste
xtr
em
e
va
lue
s
Distribution of smallest extreme
values
Determination of the
limits and the
distribution of n, and k:
Using two interpolation
points solving
Ramberg Osgood
iteratively for n and k
𝜀0,2𝑖 =𝑅𝑝0,2𝑖𝐸
+𝑅𝑝0,2𝑖𝐾
1𝑛
𝜀𝑔𝑗 =𝑅𝑚𝑗
𝐸+
𝑅𝑚𝑗
𝐾
1𝑛
σa
εaεgεg,min εg,max
Rm
Rm,min
Rm,max
ε0,2
Rp0,2
Rp0,2;min
Rp0,2;max
ε0,2,min ε0,2,max
σa
εaεgεg,min εg,max
Rm
Rm,min
Rm,max
ε0,2
Rp0,2
Rp0,2;min
Rp0,2;max
ε0,2,min ε0,2,max
Generation of
point clouds using
monte carlo
simulation with m
samplings
1
2
Loop to j, i = m
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3. Simulation process – material properties
19
Determination of the
variance of K and n using
1000 Monte Carlo
Samplings
Best fit for initial loading K
and n with Weibull
distribution
0,19650,19500,19350,19200,19050,18900,1875
180
160
140
120
100
80
60
40
20
0
n
De
nsit
y
Location 164,4
Scale 0,1952
N 1000
Weibull
Density function n
0,10400,10240,10080,09920,09760,0960
140
120
100
80
60
40
20
0
n'
De
nsit
y
Location 83,37
Scale 0,1026
N 1000
Weibull
Density function n'
0,17700,17550,17400,17250,17100,16950,16800,1665
180
160
140
120
100
80
60
40
20
0
n''
De
nsit
y
Location 143,5
Scale 0,1746
N 1000
Weibull
Density function n''
132813241320131613121308
120
100
80
60
40
20
0
K''
De
nsit
y
Location 435,0
Scale 1326
N 1000
Weibull
Density function K''
1180117611721168116411601156
140
120
100
80
60
40
20
0
K
De
nsit
y
Location 389,9
Scale 1174
N 1000
Weibull
Density function K
940936932928924920916
140
120
100
80
60
40
20
0
K'
De
nsit
y
Location 310,5
Scale 936,2
N 1000
Weibull
Density function K'
Proportionate transfer
of variance at K‘, K‘‘,
n‘ and n‘‘
Density function of n
Density function of n‘
Density function of n‘‘ Density function of K‘‘
Density function of K‘
Density function of K
K‘‘n‘‘
n‘
n K
K‘
Den
sity
De
nsity
Den
sity
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3. Simulation process
20
Loadcase simulation
Load
definition
Lifetime distribution
Carrying
Capacity
Material
properties
Werkstoff-
charakterisierung
Bauteil-
geometrie
Beanspruchungs-
Zeitfunktion
Zyklische Spannungs
Dehnungskurve
Dehnungswöhlerlinie
FEM-Analyse Schädigungsparameter
Schädigungsparameter-
wöhlerlinie
SchädigungsrechnungKlassierung des
Lastkollekivs
Bauteillebensdauer
N
εA
elastisch
plastisch
Gesamt
PSWTPB
σA, σm,
ε A
F N
N
εA
120000116000112000108000
12
10
8
6
4
2
0
Belastung
Hä
ufi
gke
it
Belastungsverteilung
La
sta
mp
litu
de
n1 n2 n3N
n1 n2 n3
La
sta
mp
litu
de
Stress / Strain
hysteresis
Definition of claim and the associated
amplitudes Characterization of the relevant
material properties
Deterioration
Stochastic Simulation for variance determination
σa
εa
P =
10 %
50 %
90 %
εa
N
Pü =
10 %50 %
90 %
10 %
50 %
90 %
Pü =
Dehnungswöhlerlinie
Stress / Strain Wöhler
curve
Identification and
consideration of non-
homogeneous loading
states
Strength
Tension Pressure Shear Bending Torsion
Characterization of the
strength model
The level of
deterioration related
stress and strain
amplitudes
Source: Haibach 2005
Source: Haibach 2005
Source: SOFEA 2015
Source: maschinenbau-wissen 2015
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3. Simulation process – deterioration
21
Ört
lich
e K
erb
sp
an
nu
ng
σ [M
Pa]
Örtliche Kerbdehnung ε [%]
First load simulation:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾
1𝑛
Correct mapping of the location
of the hysteresis
Mean stress influence
First discharge load simulation -
Masing behavior:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾′′
1𝑛′′
𝜎𝑚
Cyclic operation load
simulation:
𝜀𝑎,𝑡 =𝜎𝑎𝐸+
𝜎𝑎𝐾′
1𝑛′
Abstract representation
Formation of residual
compressive stress
Localnotc
hstr
ess σ
[MPa]
Local notch strain ε [%]
Accelerated Representative Simulation Process (ARSP)
for deterioration calculation at constant load
Nearly elastic
operating load
hysteresis
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3. Simulation process
22
Loadcase simulation
Load
definition
Lifetime distribution
Carrying
Capacity
Material
properties
Werkstoff-
charakterisierung
Bauteil-
geometrie
Beanspruchungs-
Zeitfunktion
Zyklische Spannungs
Dehnungskurve
Dehnungswöhlerlinie
FEM-Analyse Schädigungsparameter
Schädigungsparameter-
wöhlerlinie
SchädigungsrechnungKlassierung des
Lastkollekivs
Bauteillebensdauer
N
εA
elastisch
plastisch
Gesamt
PSWTPB
σA, σm,
ε A
F N
N
εA
120000116000112000108000
12
10
8
6
4
2
0
Belastung
Hä
ufi
gke
it
Belastungsverteilung
La
sta
mp
litu
de
n1 n2 n3N
n1 n2 n3
La
sta
mp
litu
de
Stress / Strain
hysteresis
Definition of claim and the associated
amplitudes Characterization of the relevant
material properties
Deterioration
Stochastic Simulation for variance determination
σa
εa
P =
10 %
50 %
90 %
εa
N
Pü =
10 %50 %
90 %
10 %
50 %
90 %
Pü =
Dehnungswöhlerlinie
Stress / Strain Wöhler
curve
Identification and
consideration of non-
homogeneous loading
states
Strength
Tension Pressure Shear Bending Torsion
Characterization of the
strength model
The level of
deterioration related
stress and strain
amplitudes
Source: Haibach 2005
Source: SOFEA 2015
Source: maschinenbau-wissen 2015Source: Haibach 2005
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σ a
N
PFailure
90 %50 %10 %
Stress
Level 1
Stress
Level 2
Endurance
Level
k
3. Simulation process – strength
23
εa
N
b2
ε'f,1ε'f,2
σ'f,2 /E
σ'f,1 /Eb1
c2c1
According to Haibach:
modeling of stress levels with
normal distribution
using Database of FKM
Determination of scattering of k:
Monte Carlo Simulation using 3
interpolation points
Determination of scattering of σ‘f , ε‘f ,
b and c: Using uniform distribution
between σ‘f1 and ε‘f1
Modeled with:
𝑁 = 𝑁𝐷 ∙𝜎
𝜎𝐷
−𝑘
Only 2 nominal literature
sources. No large database
Uniform Distribution
Scattering of
endurance strength
ND
Modeled with:
𝜀𝑎 =𝜎𝑓′
𝐸2𝑁 𝑏 + 𝜀𝑓
′ 2𝑁 𝑐
Stress Wöhler curve Strain Wöhler curve
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3. Simulation process – strength
24
-4,5-4,8-5,1-5,4-5,7-6,0-6,3
120
100
80
60
40
20
0
k
De
nsit
y
Average -5,456
Stdv 0,3371
N 1000
Normal
Density function of k
18000001600000140000012000001000000800000
100
80
60
40
20
0
ND
De
nsit
y
Average 1281073
Stdv 193278
N 1000
Normal
Density function of ND
Result for variance of k and
ND of Stress Wöhler curve
Result for variance of σ‘f , ε‘f , b and c of Strain Wöhler curve
812811810809808807806
40
30
20
10
0
Sig(f)
De
nsit
y
Density function of sigf
-0,0654-0,0660-0,0666-0,0672-0,0678-0,0684-0,0690
50
40
30
20
10
0
b
De
nsit
y
Density function of b
Compatibility condition
according to Haibach:
𝑛′ =𝑏
𝑐; 𝑘′ =
𝜎′𝑓
(𝜀′𝑓)𝑛′
Using uniform
distribution
Already
determined
Completely
determined
Density function of k
Normal
Density function of ND
Normal
ND
k
Den
sity
Den
sity
Density function of σf
Uniform
Density function of b
Uniform
b
σf
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4. DESIGN OF EXPERIMENTS
Virtual Lifetime Determination of
commercial vehicle braking systems
25
X3
X2
X1
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4. Design of Experiments
26
Spacefilling
latinhypercube sampling
Optimal cover of the
parameter space
Design Parameter
Using 3σ Normal Distribution
Material Model
Using Distribution of smallest
extreme values
Strength Model
Stress: Using Normal Distribution
Strain: Using Uniform Distribution
Load Spectra
Using Lognormal Distribution and
ARSP
Source: Siebertz 2010
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4. Design of Experiments
27
Spacefilling
latinhypercube sampling
Optimal cover of the
parameter space
Load Spectra
Using Lognormal Distribution and ARSP
1,111,081,051,020,990,96
10
8
6
4
2
0
Standardized clamping force
De
nsit
iy
Location 0,03424
Scale 0,03219
N 50
Lognormal
Densitiy function of clamping force
Ört
lich
e K
erb
sp
an
nu
ng
σ [M
Pa]
Örtliche Kerbdehnung ε [%]
Localnotc
hstr
ess σ[M
Pa]
Local notchstrain ε [%]
Ört
lich
e K
erb
sp
an
nu
ng
σ [M
Pa]
Örtliche Kerbdehnung ε [%]
Local notchstrain ε [%]
Integration of the load
spectrum through
parametric study
Higher
compressive
residual stress
Localnotc
hstr
ess σ[M
Pa]
Source: Siebertz 2010
Standardized clamping force
Distribution of clamping force
Lognormal
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Sending deterioration
outputs from FE-
Simulation
4. Design of Experiments
28
Sampling of
Design Parameter
Material model
Sampling of
Strain strength model
Calculation of lifetime
Sampling of
Stress strength model
Calculation of lifetime
Sending deterioration outputs
from FE-Simulation
Workflow
Step 1
Workflow Step 2
Wo
rkflo
w S
tep
2Workflow
Step 3
Workflow
Step 3
Sending back
results to
Robustness Analysis for
Postprocessing
Workflow
Step 4
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5. RESULT EVALUATION
Virtual Lifetime Determination of
commercial vehicle braking systems
29
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5. Result Evaluation
30
10,10,01
99,999
90807060504030
20
10
5
32
1
0,1
Standardized time to failure
Pe
rce
nt
FKM approx.
PSWT approx.
Real data
Variable
Probability plot of FKM; PSWT; Real data
Full data - ML-Estimation
Weibull
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10,10,01
99,999
90807060504030
20
10
5
32
1
0,1
Standardized time to failure
Pe
rce
nt
FKM approx.
PSWT approx.
Real data
Variable
Probability plot of FKM; PSWT; Real data
Full data - ML-Estimation
Weibull
5. Result Evaluation
31
Very good approximation
using stress based FKM
algorithm
Nearley same Weibull shape
parameter b
Small deviation in
characteristic life time T
Larger deviation on upper levels
using strain based PSWT
Pronounced Differences to conservative side
for PSWT Additionally insufficient fit
FKM approximation slightly overestimates the
real life time
Reason for this behavior:
Increasing plastic component of
strain amplitude
06
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6. SUMMARY AND OUTLOOK
Virtual Lifetime Determination of
commercial vehicle braking systems
32
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6. Summary and Outlook
Mapping of systematic uncertainties of
Design Parameter
Material model
Strength model
Load
Development of a Accelerated Representative Simulation Process
Succesfull development of a virtual lifetime distribution
Check of reproducibility
Future investigations to the influence of plastic compontents in
strain amplitudes for deterioration calculation
Investigations to the strain based strength model
33
Increase of maturity level in early design
stages!
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THANK YOU FOR YOUR
ATTENTION!
34