Failure Probability F(t) Sensitive Robust Shape parameter b STOCHASTIC SIMULATION APPROACH FOR REALISTIC LIFETIME FORECAST AND ASSURANCE OF COMMERCIAL VEHICLE BRAKING SYSTEMS Virtual Lifetime Determination Technologie Transfer Initiative 12. Weimar Optimization and Stochastic Days 2015 Durability (t-t0) 10 6 LC M. Sc. Martin Dazer Dipl.-Ing. Stefan Kemmler Prof. Dr.-Ing. Bernd Bertsche Dr.-Ing. Tobias Leopold Dipl.-Ing Jens Fricke 06.11.2015 Institute of Machine Components Reliability Engineering Technology Transfer Initiative Knorr-Bremse Group Systeme für Nutzfahrzeuge GmbH
Failure Probability F(t) Sensitive Robust Shape parameter b Research Cooperation Virtual Lifetime Determination Technologie Transfer Initiative Durability (t-t0) 10 6 LC 2
Agenda 1. Motivation 2. Application example: brake caliper 3. Simulation process 4. Design of Experiments 5. Result evaluation 6. Summary and Outlook 3
Virtual Lifetime Determination of commercial vehicle braking systems 1. MOTIVATION 4
1. Motivation Real durability tests t max Identification of specific product properties: t min Systematic and unsystematic scattering of lifetime Failure mechanism Lowest and longest cycle times Failure distribution Control of correct targetengineering by requirements Scattering produces a characteristic lifetime distribution 5
F(t) Stress Amplitude (log) Stress Amplitude (log) 1. Motivation 50% Endurable nominal load amplitude in operation Occurring nominal load amplitude on component Motivation: Realistic lifetime prediction through stochastic simulation of stress and strength Load cycles N (log) Nominal load cycles N nom (log) 90% Scatter band of endurable load amplitudes in operation 50% 10% 90% Scatter band of the load amplitudes occur on the component Stochastic Simulation 10% cumulative density Scatter band of load cycles N Source: Bertsche 2004 Load cycles N (log) 6
1. Motivation Input Variability Simulation System Output Variability Lifetime 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 7
1. Motivation FEA-Simulation model Mapping of damage-related effects Stress amplitudes Strain amplitudes Parametric CAD-Model Mapping the caliper geometry Changes in the geometry Parameter study Automatic simulation of the established DOE Statistical evaluation Source: Dynardo 2015 8
Virtual Lifetime Determination of commercial vehicle braking systems 2. APPLICATION EXAMPLE BRAKE CALIPER 9
Percent 2. Application example Brake caliper Brake caliper Piston Pressure cylinder dummy Piston Have to withstand high loads in case of overload Therefore tested with high loads Failures in Low-Cycle-Fatigue range (LCF) Probability Probability functionplot of lifetime for lifetime Weibull Weibull Full data Full ML-Estimation data - Brake disc Brake pads Percent 99 90 80 70 60 50 40 30 20 10 Statistics Shape 2,89116 Scale 0,465464 A v erage 0,414998 Stdv. 0,155908 Median 0,410044 5 Application example: Brake caliper 3 2 1 0,09 0,1 0,15 0,2 0,3 0,4 Standardized lifetime 0,5 0,6 0,7 0,8 0,9 10
Virtual Lifetime Determination of commercial vehicle braking systems 3. SIMULATION PROCESS 11
tamplitude 06.11.2015 3. Simulation process Tension Pressure Shear Bending Torsion Loadcase simulation Werkstoffcharakterisierung Zyklische Stress Spannungs / Strain Dehnungskurve hysteresis εa ela D Source: maschinenbau-wissen 2015 Definition of claim and the associated amplitudes Load definition Material properties Characterization of the relevant material properties Source: Haibach 2005 The level of deterioration related stress and strain amplitudes Stochastic Simulation Identification and consideration of nonhomogeneous loading states Source: SOFEA 2015 Deterioration σ a P = 10 % 50 % 90 % Lifetime distribution Carrying Capacity Bauteilgeometrie Strength Beanspruchungs- Zeitfunktion for variance determination ε a igkeit ε a 12 10 8 6 Pü = 10 % 50 % 90 % Klassierung des Lastkollekivs Belastungsverteilung FEM-Analyse Dehnungswöhlerlinie F σ A, σ m, ε A Stress / Strain Wöhler curve 12 Sc Schädigungsparamete εa wöhlerlinie Pü = 10 % 50 % 90 % Characterization of the strength model Source: Haibach 2005 N Schädigungsrechnung
3. Simulation process load definition Real force flow in the brake caliper Contact surface Clamping force Simulated force flow in the brake caliper Contact surface system boundary Clamping force causes high FE-calculation time System simulation to determine deterioration Same damage state system boundary Bolt pretension for lower FE-calculation time 13
3. Simulation process load definition There are variations in the clamping force occuring through hysteresis effects Load spectrum is determined from test reports Best-Fit: Lognormal distribution Probability function for clamping force Probability function of clamping force Lognormal - 95%-KI % KI Densitiy Distribution function of clamping of force force Lognormal 99 95 Test for goodness of fit Lognormal A D = 0,242 p-value = 0,759 10 8 Shape 0,03424 Scale 0,03219 N 50 Percent 80 50 20 5 Densitiy 6 4 2 1 0,95 1 1,05 1,1 Standardized clamping force Standardized clamping force 1,15 0 0,96 0,99 1,02 1,05 1,08 Standardized clamping force Standardized clamping force 1,11 14
tamplitude 06.11.2015 3. Simulation process Tension Pressure Shear Bending Torsion Loadcase simulation Werkstoffcharakterisierung Zyklische Stress Spannungs / Strain Dehnungskurve hysteresis εa ela D Source: maschinenbau-wissen 2015 Definition of claim and the associated amplitudes Load definition Material properties Characterization of the relevant material properties Source: Haibach 2005 The level of deterioration related stress and strain amplitudes Stochastic Simulation Identification and consideration of nonhomogeneous loading states Source: SOFEA 2015 Deterioration σ a P = 10 % 50 % 90 % Lifetime distribution Carrying capacity Bauteilgeometrie Strength Beanspruchungs- Zeitfunktion for variance determination ε a igkeit ε a 12 10 8 6 Pü = 10 % 50 % 90 % Klassierung des Lastkollekivs Belastungsverteilung FEM-Analyse Dehnungswöhlerlinie F σ A, σ m, ε A Stress / Strain Wöhler curve 15 Sc Schädigungsparamete εa wöhlerlinie Pü = 10 % 50 % 90 % Characterization of the strength model Source: Haibach 2005 N Schädigungsrechnung
Stress σ [Mpa] 3. Simulation process material properties 1000 900 800 700 600 500 400 300 200 100 0 Material behavior GJS-600-6 Mathematical modeling using Ramberg-Osgood-equation: ε a,t = σ a E + σ a K R e = 370 MPa R e = first load yield strength R e = cyclic yield strength 1 n R e = 500 MPa Significantly higher cyclic yield strength R e Material shows clear solidifying behavior 0 1 2 3 4 5 6 7 8 9 10 Strain ε [%] Cyclic First load First discharge General Ramberg-Osgoodequation: ε a,t = σ a E + σ a K 1 n Modelling the fist load curve: K = 1160 n = 0,19 Modelling the first discharge curve: K = 1300 n = 0,17 Modelling the cyclic curve: K = 924,9 n = 0,1 Nominal material behavior 16
Stress σ [Mpa] 3. Simulation process material properties 1000 900 Material behavior GJS-600-6 Mathematische Modellierung mit Ramberg-Osgood- Gleichung General Ramberg-Osgoodequation: ε a,t = σ a E + σ a K 1 n 800 700 600 500 400 300 200 100 R e = 500 MPa R e = 370 MPa R e = zügige Streckgrenze R e = zyklische Streckgrenze Deutlich höhere zyklische How can Streckgrenze the scattering R e behavior mapped Werkstoff zeigt deutliches mathematically? verfestigendes Verhalten Modelling the fist load curve: K = 1160 n = 0,19 Modelling the first discharge curve: K = 1300 n = 0,17 Modelling the cyclic curve: K = 924,9 n = 0,1 0 0 1 2 3 4 5 6 7 8 9 10 Strain ε [%] First load Nominal material behavior 17
Distribution of smallest extreme values 3. Simulation process material properties Only data from the static tensile test available Mathematical derivation of the scattering of n and k σ a R m,max R m 2 Determination of the limits and the distribution of n, and k: R m,min R p0,2;max R p0,2 R p0,2;min 1 Generation of point clouds using monte carlo simulation with m samplings Using two interpolation points solving Ramberg Osgood iteratively for n and k Loop to j, i = m ε 0,2i = R p0,2 i E + R p0,2 i K 1 n ε 0,2,min ε 0,2 ε 0,2,max ε g,min ε g ε g,max ε a Distribution of smallest extreme values ε gj = R m j E + R m j K 1 n 18
Density Density Density 3. Simulation process material properties Determination of the variance of K and n using 1000 Monte Carlo Samplings Density 180 160 140 120 100 80 60 40 20 Location 164,4 Scale 0,1952 N 1000 Density function of n Density function n Weibull Density 140 120 100 80 60 40 20 Location 389,9 Scale 1174 N 1000 Density function of K Density function K Weibull Best fit for initial loading K and n with Weibull distribution 0 140 120 100 0,1875 0,1890 0,1905 0,1920 0,1935 0,1950 n Location 83,37 Scale 0,1026 N 1000 n Density function of n Density function n' Weibull 0,1965 0 140 120 100 1156 1160 Location 310,5 Scale 936,2 N 1000 1164 1168 K K 1172 1176 Density function of K Density function K' Weibull 1180 Density 80 60 Density 80 60 40 40 Proportionate transfer of variance at K, K, n and n 20 0 180 160 140 0,0960 0,0976 0,0992 0,1008 0,1024 0,1040 n' Location 143,5 Scale 0,1746 N 1000 n Density function n'' Weibull Density function of n 20 0 120 100 916 920 924 928 932 936 K' Location 435,0 Scale 1326 N 1000 K Density function K'' Weibull Density function of K 940 Density 120 100 80 Density 80 60 60 40 20 40 20 0 0,1665 0,1680 0,1695 0,1710 0,1725 0,1740 0,1755 0,1770 n'' n 0 1308 1312 1316 1320 K'' K 1324 1328 19
tamplitude 06.11.2015 3. Simulation process Tension Pressure Shear Bending Torsion Loadcase simulation Werkstoffcharakterisierung Zyklische Stress Spannungs / Strain Dehnungskurve hysteresis εa ela D Source: maschinenbau-wissen 2015 Definition of claim and the associated amplitudes Load definition Material properties Characterization of the relevant material properties Source: Haibach 2005 The level of deterioration related stress and strain amplitudes Stochastic Simulation Identification and consideration of nonhomogeneous loading states Source: SOFEA 2015 Deterioration σ a P = 10 % 50 % 90 % Lifetime distribution Carrying Capacity Bauteilgeometrie Strength Beanspruchungs- Zeitfunktion for variance determination ε a igkeit ε a 12 10 8 6 Pü = 10 % 50 % 90 % Klassierung des Lastkollekivs Belastungsverteilung FEM-Analyse Dehnungswöhlerlinie F σ A, σ m, ε A Stress / Strain Wöhler curve 20 Sc Schädigungsparamete εa wöhlerlinie Pü = 10 % 50 % 90 % Characterization of the strength model Source: Haibach 2005 N Schädigungsrechnung
Örtliche Local notch Kerbspannung stress σ [MPa] σ [MPa] 3. Simulation process deterioration First load simulation: ε a,t = σ a E + σ a K 1 n Cyclic operation load simulation: ε a,t = σ a E + σ a K 1 n σ m Correct mapping of the location of the hysteresis Mean stress influence Local notch strain ε [%] Nearly elastic operating load hysteresis Örtliche Kerbdehnung ε [%] Accelerated Representative Simulation Process (ARSP) for deterioration calculation at constant load Formation of residual compressive stress First discharge load simulation - Masing behavior: ε a,t = σ a E + σ a K 1 n Abstract representation 21
tamplitude 06.11.2015 3. Simulation process Tension Pressure Shear Bending Torsion Loadcase simulation Werkstoffcharakterisierung Zyklische Stress Spannungs / Strain Dehnungskurve hysteresis εa ela D Source: maschinenbau-wissen 2015 Definition of claim and the associated amplitudes Load definition Material properties Characterization of the relevant material properties Source: Haibach 2005 The level of deterioration related stress and strain amplitudes Stochastic Simulation Identification and consideration of nonhomogeneous loading states Source: SOFEA 2015 Deterioration σ a P = 10 % 50 % 90 % Lifetime distribution Carrying Capacity Bauteilgeometrie Strength Beanspruchungs- Zeitfunktion for variance determination ε a igkeit ε a 12 10 8 6 Pü = 10 % 50 % 90 % Klassierung des Lastkollekivs Belastungsverteilung FEM-Analyse Dehnungswöhlerlinie F σ A, σ m, ε A Stress / Strain Wöhler curve 22 Sc Schädigungsparamete εa wöhlerlinie Pü = 10 % 50 % 90 % Characterization of the strength model Source: Haibach 2005 N Schädigungsrechnung
3. Simulation process strength Stress Wöhler curve Strain Wöhler curve σ a According to Haibach: modeling of stress levels with normal distribution using Database of FKM ε a ε' f,1 ε' f,2 c 1 c 2 Only 2 nominal literature sources. No large database Stress Level 1 10 % P Failure 50 % 90 % k Scattering of endurance strength N D σ' f,2 /E Uniform Distribution b 2 Stress Level 2 σ' f,1 /E b 1 Endurance Level Modeled with: N = N D σ σ D k Modeled with: ε a = σ f E 2N b + ε f 2N c N N 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 23
Density Density 3. Simulation process strength Result for variance of k and N D of Stress Wöhler curve Density function of k Normal Density function of k Normal Density function of σ f Uniform Density function of sigf Result for variance of σ f, ε f, b and c of Strain Wöhler curve 120 Average -5,456 Stdv 400,3371 N 1000 100 Density 80 60 40 20 0-6,3-6,0-5,7-5,4 k k -5,1-4,8-4,5 Density 30 20 10 0 806 807 808 809 Sig(f) σ f 810 811 812 Compatibility condition according to Haibach: Already determined Using uniform distribution 100 Density function of N D Normal Density function of ND Normal Average 1281073 50 Stdv 193278 N 1000 Density function of b Density Uniform function of b n = b c ; k = σ f (ε f ) n 80 40 Density 60 40 Density 30 20 Completely determined 20 10 0 800000 1000000 1200000 1400000 1600000 1800000 ND N D 0-0,0690-0,0684-0,0678-0,0672-0,0666 b b -0,0660-0,0654 24
X 1 X 3 X 2 Virtual Lifetime Determination of commercial vehicle braking systems 4. DESIGN OF EXPERIMENTS 25
4. Design of Experiments Spacefilling latinhypercube sampling Source: Siebertz 2010 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 26
Local Örtliche notch Kerbspannung stress σ [MPa] Local Örtliche Kerbspannung notch stress σ [MPa] σ [MPa] 4. Design of Experiments Spacefilling latinhypercube sampling Load Spectra Using Lognormal Distribution and ARSP 10 Distribution Densitiy function of clamping of clamping force force Lognormal Location 0,03424 Scale 0,03219 N 50 Higher compressive residual stress Local notch strain ε [%] 8 Densitiy 6 4 Örtliche Kerbdehnung ε [%] 2 Source: Siebertz 2010 0 0,96 0,99 1,02 1,05 1,08 Standardized clamping force Standardized clamping force 1,11 Optimal cover of the parameter space Integration of the load spectrum through parametric study Örtliche Kerbdehnung ε [%] Local notch strain ε [%] 27
Workflow Step 2 4. Design of Experiments Sampling of Design Parameter Material model Workflow Step 1 Sending deterioration outputs from FE-Simulation Workflow Step 2 Sampling of Stress strength model Calculation of lifetime Workflow Step 3 Sending deterioration outputs from FE- Simulation Sampling of Strain strength model Calculation of lifetime Workflow Step 3 Workflow Step 4 Sending back results to Robustness Analysis for Postprocessing 28
Virtual Lifetime Determination of commercial vehicle braking systems 5. RESULT EVALUATION 29
5. Result Evaluation Probability plot of FKM; PSWT; Real data Weibull Full data - ML-Estimation Percent 99,9 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Variable FKM approx. PSWT approx. Real data 0,1 0,01 0,1 Standardized time to failure 1 30
5. Result Evaluation Percent 99,9 99 90 80 70 60 50 40 30 20 10 5 3 2 1 Probability plot of FKM; PSWT; Real data Weibull Full data - ML-Estimation Variable FKM approx. PSWT approx. Real data 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 0,1 0,01 0,1 Standardized time to failure 1 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 31
Virtual Lifetime Determination of commercial vehicle braking systems 6. SUMMARY AND OUTLOOK 32
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 Increase of maturity level in early design stages! 33
martin.dazer@ima.uni-stuttgart.de martin.dazer@knorr-bremse.com THANK YOU FOR YOUR ATTENTION! 34