1. CTU in Prague, Faculty of Mechanical Engineering DAF Page 1 Mean stress effect Smith‘s diagram  FL

2. CTU in Prague, Faculty of Mechanical Engineering DAF Page 2 Haigh’s diagram k=1 and we obtain Goodman straight line, k=2 andGerber parabola, k=1 andSoderberg brittle material approximation.   A  m  F R e  C, k = 2 k = 1 Mean stress effect  FL ff

3. CTU in Prague, Faculty of Mechanical Engineering DAF Page 3 Mean stress effect

4. 1.CTU in Prague, Faculty of Mechanical Engineering 1.DAF1. 1.Page 4 Safety factor of unlimited fatigue life (permanent strength) 1.Alternating stress (R=-1) operational loading stress amplitude  a fatigue limit of the real part in the critical cross section area  FL,N

5. 1.CTU in Prague, Faculty of Mechanical Engineering 1.DAF1. 1.Page 5 Safety factor of unlimited fatigue life (permanent strength) 2.Pulsating stress operational loading stress amplitude  a operational mean stress  m fatigue limit of the real part in the critical cross section area  FL,N AA mm ReRe AA mm MM aa P (2) (1) ReRe M M´ (3) (1) (2) In the point M

6. CTU in Prague, Faculty of Mechanical Engineering DAF Page 6 AA AA aa P (2) (1) ReRe (2) M M´ (3) Point M Safety factor of unlimited fatigue life (permanent strength) mm ReRe mm MM

7. CTU in Prague, Faculty of Mechanical Engineering DAF Page 7 AA mm ReRe AA mm MM aa P (3) (2) (1) ReRe M M´ In the point M´ (3) Result Safety factor of unlimited fatigue life (permanent strength)

8. CTU in Prague, Faculty of Mechanical Engineering DAF Page 8 Safety factor of unlimited fatigue life (permanent strength) 3.Bending and torsion fatigue alternating loading operational bending stress amplitude  a operational torsion stress amplitude  a normal stress fatigue limit in the critical cross section area  FL shear stress fatigue limit in the critical cross section area  FL

9. CTU in Prague, Faculty of Mechanical Engineering DAF Page 9  FL    FL k=2 AA AA M aa aa P (2) (1) (1) ellipse (2) In the point M Safety factor of unlimited fatigue life (permanent strength)

10. CTU in Prague, Faculty of Mechanical Engineering DAF Page 10 mm aa mm aa       aa mm Safety factor of unlimited fatigue life (permanent strength)

11. statistic quantile: Example  a =280 MPa, s FL =38 MPa  FL =420 MPa variability v a =12% Bezpečnost MS EXCEL: =NORMSDIST() =NORMSINV() Probability of fracture P p Probability of fracture P statistic distribution Mean value of the fatigue limit Mean value of the amplitude safety factor n C S FL SaSa  FL aa

12. CTU in Prague, Faculty of Mechanical Engineering DAF Page 12 Equivalent alternating stress amplitude AA mm ff ReRe k=1 aa mm M  a,, eq

13. CTU in Prague, Faculty of Mechanical Engineering DAF Page 13 Dynamic loadings The type of loadingPercent of operation Sinusoidal with constant Aaplitude2 Triangular or trapezoidal time Hystory with constant amplitude 5 Block loading with constant amplitude in every blocs 12 Stationary random process9 Non-stationary random process13 Staionary in Patrs41 Transients processes13 Others and special processes5 DYNAMIC LOAD (Time History) Deterministic Stochastic Non-PeriodicPeriodic Transient Harmonic Nearly Periodic Alternativ Per. StationaryUnsteady Non-Ergodic Stationary in Parts. Ergodic NarrowbandWideband Reording x 1 (t) Reording x 2 (t) Reording x 3 (t) Reording x 4 (t) Along Realization Over Realization Set

14. CTU in Prague, Faculty of Mechanical Engineering DAF Page 14 Rain-flow method Mean Value Amplitude

15. CTU in Prague, Faculty of Mechanical Engineering DAF Page 15 Rain-flow matrix, two parametric counting amplitude mean value Number of loading cycles n i (frequency)

16. CTU in Prague, Faculty of Mechanical Engineering DAF Page 16 One parametric loading spectrum (histogram) Cumulative frequency h i Class i Approximation of loading spectra

17. CTU in Prague, Faculty of Mechanical Engineering DAF Page 17 Examples of Multi body simulation – dynamic loading prediction

18. CTU in Prague, Faculty of Mechanical Engineering DAF Page 18 Examples of aircraft loading spectra Start Fly-Up Horizontal Flight Fly-Down Landing Reverse of Engine Land travel, turns, braking Stress time Flight Definition relative composatio n  %  acrobatics22,4 Fight training17,8 Group flighs9,6 Instrum. flights25,6 Cross-country18,2 circuits6,4 sum100,0

19. CTU in Prague, Faculty of Mechanical Engineering DAF Page 19 Linear fatigue accumulation Miner‘s rule (Miner, 1945) Fatigue damage Limited fatigue life, Damage accumulation  FL nini NiNi

20. CTU in Prague, Faculty of Mechanical Engineering DAF Page 20 Limited fatigue life prediction The critical value of damage The range of the spectrum (number of loading cycles) Number of the spectrum repeatings to failure The mean fatigue life (probability of failure 50%) The range of the spectrum (number of operational houres, number of kilometers)

21. CTU in Prague, Faculty of Mechanical Engineering DAF Page 21 Safety life life Statistical frequency Mean life Safety life Shift according to the safety factor n L s log N s log n The safety fatigue life - probability of failure P<< 0 (for example 0,001%..... 0,00001%) The standard deviation of the fatigue life on the S-N curve s log N The standard deviation of the loading cycles s log n The total safety factor of the fatigue life n L The safety fatigue life The safety factor of the S-N curve n N(3.0... 6.0) The safety factor of the loading specra n n(1.5 … 2,.0)

22. CTU in Prague, Faculty of Mechanical Engineering DAF Page 22 Probability of fracture PfPf life Statistical frequency Mean life Safety life Shift according to the safety factor n L s log N s log n Assumption: log-normal fatigue life probability distribution Calculation of statistic quantile and failure probability P f [%]

23. CTU in Prague, Faculty of Mechanical Engineering DAF Page 23 Estimate the fatigue life safety factor n L of the railway axle. The probability of the fracture should be less then 10 -4. Input data: The S-N curve of the axle steel The fatigue limit in the notch The standard deviation of the fatigue life on the S-N curve The histogram of the nominal stresses (for 5000 km) The standard deviation of the loading cycles A Example – The fatigue life prediction

24. CTU in Prague, Faculty of Mechanical Engineering DAF Page 24 R m =S ult N FL =10 6 NRNR  FL  FL,N Example – Continue Solution Estimation of the notch S-N curve (see lecture 2) factorkvalue loadingkLkL 1.00 surface finishk SF 0.67 size factorkSkS 0.70 size factorkT1.00 All stress amlitudes are higher as the fatigue limit. The Limited life must bee taken in the account

25. CTU in Prague, Faculty of Mechanical Engineering DAF Page 25 Fatigue damage  FL Example – Continue Number of the spectrum repeatings to failure is The mean fatigue life (probability of failure 50%)

26. CTU in Prague, Faculty of Mechanical Engineering DAF Page 26 R m =S ult N FL =10 6 N R =calculate  FL  FL,N Notices to the home work Solution we know S-N curve, parameters  f ‘, b, fatigue limit  FL static strength S ult. (=points S and F) we can calculate N R for S ult. we know fatigue limit of the notched part  FL,N, and point N R for S ult. we can calculate new parameters of the S-N curve of the notched part b N (=points S and F N )  a =  f ( N) b  a =  f ‘( N) b Estimation of the notch S-N curve:  FL,N =  f N ‘( N) bN S FNFN F

27. CTU in Prague, Faculty of Mechanical Engineering DAF Page 27 R m =S ult N 50% =calculate  FL  FL,N Notices to the home work – Continue Solution we have calculated parameters of the part in the critical cross- section:  fN ‘, b N, we have calculated nominal stress amplitude  a in the critical cross-section we can calculete the mean fatigue life (with the probability of failure 50%) we can calculete the safe fatigue life now we can calculate the failure probability at the end of this life  a =  f ( N) b  a =  fN ‘( N) bN Prediction of the fatigue life: S  a,nom P f [%]

28. CTU in Prague, Faculty of Mechanical Engineering DAF Page 28 PfPf life Statistical frequency Mean life Safety life Shift according to safety factors n L S log N S log n The statistic quantile Example – Continue

29. CTU in Prague, Faculty of Mechanical Engineering DAF Page 29 Questions and problems III

30. CTU in Prague, Faculty of Mechanical Engineering DAF Page 30 Questions and problems III