Fatigue Design of Welded Structures - Physical Modeling vs. Empirical Rules

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1 Fatigue Design of Welded Structures - Physical Modeling vs. Empirical Rules G. Glinka,, Ph.D., D.Sc. University of Waterloo, Department of Mechanical and Mechatronics Engineering, 2 University Ave. W, Waterloo, Ontario, Canada N2L 3G1 Phone: ext gggreg@mecheng1.uwaterloo.ca Stockholm, 21-22, October Grzegorz Glinka. All rights reserved. 1

2 Contents 1) Brief review of American and European rules concerning static strength analysis of weldments, The structural nature of welded joints Static strength of weldments The customary US/American method (AWS) The IIW/ISO method Simple welded joint analysis Example 2) Outline of contemporary fatigue analysis methods similarities, differences and limitations, the classical nominal stress method (S - N), the local strain-life approach,(ε N) the fracture mechanics approach (da/dn ΔK), 3) Load/stress histories methods of assembling characteristic load/stress histories, the rainflow counting method live example, extraction of load/stress cycles from the frequency domain data, 28 Grzegorz Glinka. All rights reserved. 2

3 4) Loads and stresses in fatigue the stress concentration in weldments, the nominal stress, the hot spot stress and the peak stress at the weld toe, the use of the Finite Element method for the stress analysis in weldments, residual stresses; 5) The strain-life (ε N) approach to fatigue life assessment of weldments material properties and their scatter, linear elastic stress distributions in weldments, the actual elastic-plastic stress-strain response at the weld toe, the multiaxial Neuber and the ESED method (uniaxial and multi-axial), superposition of applied and residual stresses, fatigue damage accumulation, deterministic fatigue life prediction the effect of the weld geometry and residual stresses, quantification of the scatter of the load, weld geometry and material properties on the on the scatter of the predicted life, application of the Monte-Carlo simulation method for the estimation of the fatigue reliability; 28 Grzegorz Glinka. All rights reserved. 3

4 6) The Fracture Mechanics approach basics of fatigue crack growth analysis stress intensity factors for cracks in weldments, the weight function approach 2-D and 3-D solutions, the residual stress effect, modeling of the fatigue crack growth of irregular planar cracks, quantification of the scatter and the Monte Carlo simulation), 7) Simple methods for improvement of fatigue performance of welded structures decrease of the stress concentration, elimination of the local bending effect, modification of the stress path, optimization of the global geometry of a welded structure, introduction of favorable residual stresses live examples 8) Recent developments in the fatigue crack growth analysis The UniGrow fatigue crack growth model for spectrum loading 28 Grzegorz Glinka. All rights reserved. 4

5 Bibliography 1. Bannantine, J., Corner, Handrock, Fundamentals of Metal Fatigue Analysis, Prentice-Hall, (complete, correct, up to date) 2. Dowling, N., Mechanical Behaviour of Materials, Prentice Hall, ( middle chapters are a great overview of most recent views on fatigue analysis) 3. M. Janssen, J. Zudeima, R.J.H. Wanhill, Fracture Mechanics, VSSD, The Netherlands, 26 (understandable, rigorous, mechanics perspective) 4. Stephens, R.I., Fatemi, A., Stephens, A.A., Fuchs, H.O., Metal Fatigue in Engineering, John Wiley, (good general reference, well done) 5. Radaj, D., Design and Analysis of Fatigue Resistant Structures, Halsted Press, (Complete, civil engineering analysis perspective) 6.Suresh, S., Fatigue of Materials, Cambridge University Press, (Encyclopedic, up to date, fundamental metallurgical perspective) 7. Anderson, T.L., Fracture Mechanics-Fundamentals and Applications, CRC Press, Boca Raton, Collins, J. A., Failure of Materials in Mechanical Design, John Wiley & Sons, New York, Broek, D., Elementary Engineering Fracture Mechanics, Martinus Nijhoff, The Hague, Hertzberg, R.W., Deformation and Fracture Mechanics of Engineering Materials, John Wiley, New York, Grzegorz Glinka. All rights reserved. 5

6 Bibliography (continued) 11. K. Iida and T. Uemura, Stress Concentration Factor Formulas Widely Used in Japan, Document IIW XIII , The International Welding Institute, Bahram Farhmand, G. Bockrath and J. Glassco, Fatigue and Fracture Mechanics of High Risk Parts, Sandor, R.J., Principles of Fracture Mechanics, Prentice Hall, Upper Saddle River, Socie, D.F., and Marquis, G.B., Multiaxial Fatigue, Society of Automotive Engineers, Inc., Warrendale, PA, Serensen, S.V., Kogayev, V.P., Shneyderovich, R.M., Nesushchaya Sposobnost i Raschyoty Detaley Machin na Prochnost, Mashinostroyene, Moskva, 1975 (in Russian) 16. Machutov, N.A., Resistance of Structural Elements Against Brittle Fracture, Mashinostroenye, Moscow, 1973 (in Russian). 17. Cherepanov, G.P., Mechanika chrupkovo razrusheniya, Nauka, Moscow, 1974 (in Rusian). 18. Bathias, C., and Bailon, J.-P., La Fatigue des Materiaux et des Structures, Maloine S.A. Editeur, Paris, 198 (in French). 19. Haibach, E., Betriebsfestigkeit, VDI Verlag, Dusseldorf, 1989 (in German). 2. Barsom, J.M, and Rolfe, S.T., Fracture and Fatigue Control in Structures, Prentice Hall, New Jersey, Courtney, T.H., Mechanical Behavior of Materials, McGraw Hill, New York, Grzegorz Glinka. All rights reserved. 6

7 Bibliography (continued) 22. Collins, J.C., Mechanical Design of Machine Elements and Machines, John Wiley & Sons, New York, Shigley, J.E., and Mischke, C.R., Mechanical Engineering Design, McGraw-Hill, New York, Juvinal, R.C., and Marshek, K.M., Fundamentals of Machine Components Design, John Wiley & Sons, New York, Norton, R.L., Machine Design An Integrated Approach, Prentice Hall, New Jersey, Hamrock, B.J., Jacobson, Bo and Schmid, S.R., Fundamentals of Machine Elements, McGraw- Hill, Boston, Orthwein, W., Machine Component Design, West Publishing Company, New York, J.-L. Fayard, A. Bignonnet and K. Dang Van, Fatigue Design Criterion for Welded Structures, Fatigue and Fracture of Engineering Materials and Structures, vol. 19, No.6, 19996, pp V.A. Ryakhin and G.N. Moshkarev, Durability and Stability of Welded Structures in Earth Moving Machinery, Mashinostroenie, Moscow, 1984 (in Russian) V. I. Trufyakov (editor), The Strength of Welded Joints under Cyclic Loading, Naukova Dumka, Kiev, 199 (in Russian). 31. J.Y. Young and F.V. Lawrence, Analytical and Graphical Aids for the Fatigue Design of Weldments, Fracture and Fatigue of Engineering Materials and Structures, vol. 8, No.3, 1985, pp Grzegorz Glinka. All rights reserved. 7

8 Bibliography (continued) 32. S.V. Petinov, Principles of Engineering Fatigue Analyses of Ship Structures, Sudostroyenie, Leningrad, 199 (in Russian and in English) 33. C.C. Monahan, Early Fatigue Cracks Growth at Welds, Computational Mechanics Publications, Southampton UK, Y. Murakami et. al, (editor), Stress Intensity Factors Handbook, Pergamon Press, Oxford, O.W. Blodgett, Design of Welded Structures, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, A. Almar-Naess, Fatigue Handbook Offshore Steel Structures, Tapir Publishers, Trondheim, Norway, E. Niemi, Structural Stress Approach To Fatigue Analysis Of Welded Components, The International Institute of Welding, Doc. XIII-WG3-6-99, D. Pingsha, A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints, International Journal of Fatigue, vol. 23, 21, p R.E. Little andj.c. Ekvall, Statistical Analysis of Fatigue Data, ASTM STP 744, J.D. Burke and F.V. Lawrence, The Effect of Residual Stresses on Fatigue Life, FCP Report no. 29, University of Illinois, College of Engineering, J.G. Hicks, Welded Joint Design, Abington Publishing, UK, 1997-edition 2 and 1999-edition Grzegorz Glinka. All rights reserved. 8

9 Relevant Standards Technical Report on Fatigue Properties - SAE J199, SAE Book of Standards, 1992, SAE, p Standard Practices for Cycle Counting in Fatigue Analysis, ASTM Standard E , ASTM Book of Standards, vol. 3.1, 1994, p. 789 Standard Recommended Practice for Contant-Amplitude Low-Cycle Fatigue Testing, ASTM Standard E 66-8, ASTM Book of Standards, vol. 3.1, 1994, p. 69 Standard Practice for Stastical Analysis of Linearized Stress-Life (S-N) and Strain-Life (ε-n) Fatigue Data, ASTM Standard E , ASTM Book of Standards, vol. 3.1, 1994, p. 682 Standard Test Method for Measurement of Fatigue Crack Growth Rates, ASTM Standard E , ASTM Book of Standards, vol. 3.1, Grzegorz Glinka. All rights reserved. 9

10 Relevant websites - Fatigue (**Good basic course notes and tutorials on Fracture Mechanics) (*good overview notes) (*Failure of Materials, examples) (*General notes on fatigue) (*Tutorials on Strength theories, Stress concentration factors and Pressure vessel design) (*Calculator for shaft design, static and fatigue) (*Excel Files, stress concentration factors for shafts and design of shafts) (*Interesting notes on shaft design and Tables) (Stress concentration factors, fatigue methods) (*Fatigue software, manual and theory of fatigue methods) 28 Grzegorz Glinka. All rights reserved. 1

11 Relevant Websites Fracture Mechanics (**Good basic course notes and tutorials on Fracture Mechanics) (*Cracking of dams, history of FM, animations) (**Good notes on FM from Australia) (**Good complete notes on Fracture Mechanics) (**Concise notes/explanations of basics of FM) (* Fracture Mechanics the Wikipedia way) (**Plastic, brittle, fatigue fractures notes) Factors%22 (**NASA practical use of Fracture Mechanics good examples) a_kin/ (Catalogue of SIF) 28 Grzegorz Glinka. All rights reserved. 11

12 1) Brief review of American and European rules concerning static strength analysis of weldments, The structural nature of welded joints Static strength of weldments The customary US/American method (AWS) The IIW/ISO method Simple welded joint analysis Example 28 Grzegorz Glinka. All rights reserved. 12

13 When permanent joints are an appropriate design solution, welding is often an economically attractive alternative to threaded joints. Most industrial welding processes involve local fusion of the parts to be joined, at their common interfaces, to produce a weldment. In the design of welded joints in a structure, two most common types of welds are used, i.e. butt welds and fillet welds. A butt weld is usually used to join two plates of the same thickness and is considered to be an integral part of the loaded component. Calculations are carried out as if stresses and trains in the weld and in the base metal were the same. The influence of the weld reinforcement (overfill) is ignored. Fillet welds are non-integral in character, and have such a shape and orientation relative to the loading that it almost disallows application of simple stress analysis. Conventional practice in welding engineering design has always been to quantify the size of the weld depending on the stress acting in the weld throat cross sectional area. Thus, fillet weld sizes are determined by reference to allowable shear stress in the throat cross section. Fillet welds being parallel to the direction of the loading force are called longitudinal fillet welds. Fillet welds normal to the direction of the loading force are called transverse fillet welds. The stress state in the weld throat plane consists in general of three stress components, i.e. two shear components τ 1 and τ 2 and one normal stress component Grzegorz Glinka. All rights reserved. 13

19 Stress concentration & stress distributions in weldments peak r E n hs D n P F B peak t C A C M Various stress distributions in a butt weldment; Normal stress distribution in the weld throat plane (A), Through the thickness normal stress distribution in the weld toe plane (B), Through the thickness normal stress distribution away from the weld (C), Normal stress distribution along the surface of the plate (D), Normal stress distribution along the surface of the weld (E), Linearized normal stress distribution in the weld toe plane (F). 28 Grzegorz Glinka. All rights reserved. 19

21 Stress concentration & stress distributions in weldments t 1 peak r Θ E hs D P M t A B F C Various stress distributions in a T-butt weldment with transverse fillet welds; n C Normal stress distribution in the weld throat plane (A), Through the thickness normal stress distribution in the weld toe plane (B), Through the thickness normal stress distribution away from the weld (C), Normal stress distribution along the surface of the plate (D), Normal stress distribution along the surface of the weld (E), Linearized normal stress distribution in the weld toe plane (F). 28 Grzegorz Glinka. All rights reserved. 21

22 Static strength analysis of weldments The static strength analysis of weldments requires the determination of stresses in the load carrying welds. The throat weld cross section is considered to be the critical section and average normal and shear stresses are used for the assessment of the strength under axial, bending and torsion modes of loading. The normal and shear stresses induced by axial forces and bending moments are averaged over the entire throat cross section carrying the load. The maximum shear stress generated in the weld throat cross section by a torque is averaged at specific locations only over the throat thickness but not over the entire weld throat cross section area. Non-load carrying welds Load carrying welds 28 Grzegorz Glinka. All rights reserved. 22

24 Stress components in the weld throat cross section plane in a T- butt weldment with load-carrying transverse fillet welds (nominal stress components in the weld throat cross section!!) = P cosα/a; τ 1 = P sinα/a ; τ 2 = R/A A = L t = 2L h sinα P R τ 1 τ 2 L t α n R P 28 Grzegorz Glinka. All rights reserved. 24 h

25 Stress components in the weld throat cross section plane in a T- butt weldment with load - carrying transverse fillet welds (simplified combination of stresses in the weld throat cross section according to the customary American method!!) =!!; τ 1 = X =P/A; τ 2 =R/A Calculation of stresses in a fillet weld P R (It is assumed that only shear stresses act in the weld throat!) A = L t = 2L h sinα τ 1 τ 2 x L t α n R P 28 Grzegorz Glinka. All rights reserved. 25

26 Stress components in the weld throat cross section plane in a T- butt weldment with load-carrying transverse fillet welds (the customary American method!!) Calculation of stresses in a fillet weld The resultant shear stress Equivalent stress ( ) ( τ ) τ τ = + P R eq = 3τ ( τ ) ( τ ) = τ 1 τ τ 2 L t α n R P 28 Grzegorz Glinka. All rights reserved. 26

27 a) Fillet welds under primary shear and bending load b V b) V V V = tb τ = 1, V V shear α d V d) M t V h l V c) M V τ M 1, M = Mc I = M d V l 2 tbd 2 V l tbd = = 2 bending M M M V 28 Grzegorz Glinka. All rights reserved. 27

28 Fillet welds in primary shear and bending: the American customary method of combining the primary shear and bending stresses (according to R.C.Juvinal & K.M. Marshek in Fundamentals of Machine Component Design, Wiley, 2) ) V τ 1 = τ 1 = τ τ = + = V M 1, V 1, M M V τ 2 2 V V l V 1 1 tb tbd tb = + = + l d 2 2 d) Acceptable design: M τ ys 1 3 V τ 1 = V tb 1+ l d ys 3 or V tb l 3 1+ d 2 ys 28 Grzegorz Glinka. All rights reserved. 28

29 Fillet welds in primary shear and bending: the ISO/IIW method of combining the primary shear and bending shear stresses a) Vn, V τ Vt, τ b) M.n M V τ M.t M V eq V = V tb M Vn, Mn, Vt, Mt, = + + τ τ ys M = Mc I d V l 2 tbd 2 V l tbd = = 2 for α = eq = V 21 l l + V V M M = + tb d d 28 Grzegorz Glinka. All rights reserved ys

30 Static Strength Assessment of Fillet Welds The AWS method: It is assumed that the weld throat is in shear for all types of load and the shear stress in the weld throat is equal to the normal stress induced by bending moment and/or the normal force and to the shear stress induced by the shear force and/or the torque. There can be only two shear stress components acting in the throat plane - namely τ 1 and τ 2. Therefore the resultant shear stress can be determined as: τ = τ + τ The weld is acceptable if : τ < τ = ys ys 3 Where: τ ys is the shear yield strength of the: weld metal for fillet welds and parent metal for butt welds 28 Grzegorz Glinka. All rights reserved. 3

31 Stress components in the weld throat cross section plane in a T- butt weldment with load-carrying transverse fillet welds (correct solid mechanics combination of stresses in the weld throat the European approach) = Pcosα/A τ 1 = Pcosα/A τ 2 = R/A P R A = 2L t cosα τ 1 Resultant equivalent stress ( ) ( ) eq = + 3 τ1 + τ2 τ 2 L t α n R P 28 Grzegorz Glinka. All rights reserved. 31

32 The IIW/ISO Method: International Welding Institute employs a method in which the stresses are resolved into three components across the weld throat. These components of the stress tensor are the normal stress component perpendicular to the throat plane, the shear stress τ 2 acting in the throat parallel to the axis of the weld and a shear stress component τ 1 acting in the throat plane and being perpendicular to the longitudinal axis of the weld. The proposed formula for calculating the equivalent stress is: ( ) = β + 3 τ + τ eq Static strength assessment of fillet welds The weld is acceptable if : eq < ys Where: ys is the yield strength of the: weld metal for fillet welds and parent metal for butt welds The β coefficient is accounting for the fact that fillet welds are slightly stronger that it is suggested by the equivalent stress eq. β =.7 for steel material with the yield strength ys < 24 MPa β =.8 for steel material with the yield strength 24 < ys < 28 MPa β =.85 for steel material with the yield strength 28 < ys < 34 MPa β = 1 for steel material with the yield strength 34 < ys < 4 MPa 28 Grzegorz Glinka. All rights reserved. 32

33 Idealization of welds in a T-T butt welded joint; a) geometry and loadings, b) and c) position of weld lines in the model for calculating stresses ses under axial, torsion and bending loads M b T r a) b T r P d b) b P M b Weld line h 2c t c) 2c r P P = ; b = 2td M I b w c r r Weld line d b 2c τ T r Tr r d b = ; c = or J Grzegorz Glinka. All rights reserved. 33 w 2c

34 Combination of stress components induced by multiple loading modes a) A w b) c) y τ(x,y) y y r r CG x (x,y) z A p T r x P z M V ( ) 2 3( ) 2 β τ τ eqv = M + P + T + V yield IIW r = P P A w Tr = M r max τ T = r V J w, CG ( ) ys τ = τ + τ + + < τ r 3 M c I wx, x y VQ τ = It T V P M yield AWS 28 Grzegorz Glinka. All rights reserved. 34

35 EXAMPLE: Transverse fillet weld under axial loading AWS P/2 P τ1 = x = 2lt P = 2lhcosθ P = 2lhcos45 P = 1.414lh P/2 h P/2 t h l x θ P x eq = 3τ = 1 τ 1 P/2 3P 1.414lh P lh 28 Grzegorz Glinka. All rights reserved. 35

36 t P/2 τ 1 IIW / ISO 1 θ P/2 Psinθ Psin45 P 1 = x sinθ = = = 2lt 2lhcos45 2lh Pcosθ Pcosθ P τ1 = x cosθ = = = 2lt 2lhcosθ 2lh eq P P = β 1 + 3τ1 = β 3 2lh + = 2lh P β lh 28 Grzegorz Glinka. All rights reserved. 36

37 2) Outline of contemporary fatigue analysis methods similarities, differences and limitations, the classical nominal stress method (S - N), the local strain-life approach,(ε N) the fracture mechanics approach (da/dn ΔK), 28 Grzegorz Glinka. All rights reserved. 37

38 The Most Popular Methods for Fatigue Life Predictions Stress-Life Method or the S - N approach; uses the nominal or simple engineering stress S to quantify fatigue damage Strain-Life Method or the ε - N approach; uses the local notch tip strains ε a and stresses a to quantify the fatigue damage Fracture Mechanics or the da/dn ΔK; uses the stress intensity factor range ΔK to quatify the fatigue crack growth rate da/dn 28 Grzegorz Glinka. All rights reserved. 38

39 Information Path for Strength and Fatigue Life Analysis Material Properties Component Geometry Loading History Stress-Strain Analysis Damage Analysis Fatigue Life 28 Grzegorz Glinka. All rights reserved. 39

40 Stress Parameters Used in Static Strength and Fatigue Analyses S K t = peak n M K = S π a Y S Stress peak n n Stress peak Stress = K 2π a ρ y ρ y ρ a y d n T d n T T a) b) c) S 28 Grzegorz Glinka. All rights reserved. 4

Generation of fatigue the S-N data under rotational bending + n,max n,max Stress

42 The classical fatigue S-N curve Low cycle fatigue Part 1 High cycle fatigue Part 2 Infinite life Part 3 14 S u 12 Stress amplitude, S a (ksi) S e S y S Number of cycles, N Fully reversed axial S-N curve for AISI 413 steel. Note the break at the LCF/HCF transition and the endurance limit. 28 Grzegorz Glinka. All rights reserved. 42 S = C N = 1 N a m A m

43 The Strain-life and the Cyclic Stress-Strain Strain Curve Obtained from Smooth Cylindrical Specimens Tested Under Strain Control (Uni-axial Stress State) ε ε ε ε f Strain - Life Curve Stress -Strain Curve log (Δε/2) f /E b c e /E E log (2N f ) 2N e 1 ' ' b c n Δ ε f 2N ' = ε 2N 2 E f + f f ε = + E K ' ε 28 Grzegorz Glinka. All rights reserved. 43

44 Determination of the Complete Fatigue Crack Growth Curve (da/dn- K K and K th ) n a) c) d) b) W 2a W P or n Time K = πa Y n Log(da/dN) Applicability of Paris equation m a P K = P π a Y ΔK th Log(ΔK) K max = K c P a & b) specimens used to generate fatigue crack growth curves, c) stress history, d) fatigue crack growth rate curve. 28 Grzegorz Glinka. All rights reserved. 44

45 Information path for fatigue life estimation based on the S-N method F LOADING t GEOMETRY, K f PSO MATERIAL E ε Stress-Strain Analysis Damage Analysis Δ n MATERIAL e N o N Fatigue Life 28 Grzegorz Glinka. All rights reserved. 45

46 Steps in Fatigue Life Prediction Procedure Based on the S-N Approach The S N method a) Structure b) Component H e) Standard S-N curves c) Section with welded joint Q F Stress amplitude, Δ n /2 or Δ hs /2 K K1 K2 K3 K4 K5 V R P Weld d) Standard welded joints Number of cycles, N 28 Grzegorz Glinka. All rights reserved. 46 5

47 Steps in Fatigue Life Prediction Procedure Based on the S-N Approach (continued) n Stress f) K5 Stress, n t g) Δ 3 Δ 4 h) Fatigue damage: D D D D D 5 1 = = N 1 = = N 1 = = N 1 = = N 1 = = N ( Δ ) C ( Δ ) C ( Δ ) C ( Δ ) C ( Δ ) C i) Total damage: m 5 m m m m 5 Δ 1 t D = D + D + D + D + D ; j) Fatigue life: N blck =1/D Δ 2 Δ 5 28 Grzegorz Glinka. All rights reserved. 47

48 The Similitude Concept in the S-N Method a) Structure H Q F K 5 n Stress amplitude, Δ n /2 or Δ hs /2 K K1 K2 K3 K4 K5 Number of cycles, N The Similitude Concept states that if the nominal stress histories in the structure and in the test specimen are the same, then the fatigue response in each case will also be the same and can be described by the generic S-N curve. It is assumed that such an approach accounts also for the stress concentration, loading sequence effects, manufacturing etc. 28 Grzegorz Glinka. All rights reserved. 48

49 Information path for fatigue life estimation based on the ε-n method F LOADING GEOMETRY, K f MATERIAL t PSO E ε Stress-Strain Analysis Damage Analysis ½ Δε MATERIAL 2N f Fatigue Life 28 Grzegorz Glinka. All rights reserved. 49

51 Steps in fatigue life prediction procedure based on the ε-n approach ,7' 5,5' 2,2' e) ε Δ 3 1,1' ε log (Δε/2) f /E e /E ε f Δ peak 1 f) 2 ( ) 2 peak Neuber : = ε E ' Δ ε f = + 2 E ( 2 b ' N ) ( 2 f ε f N f ) ' c t ' f ' f Fatigue damage: D = ; D = ; D = ; D = ; N N N N Total damage: ε D = D + D + D + D ; ' n = + ' E (, ) Δε μ N f Δε ( ', ' ) f f ( ', ' ε ) f ε f μ ε μ (?,?) Δε K ε Δε p Δε e = Δ/Ε Δε 2N log (2N f ) 28 Grzegorz Glinka. All rights reserved. 51 2N e Fatigue life: N blck =1/D

notch tip and the strain history in the test specimen are the same, then the fatigue response in the notch tip region and in the

52 The Similitude Concept in the ε N Method ε peak a) Specimen x ε peak log (Δε/2) ε f ' j) Δ ε f b = ' ( 2N f ) +ε f ( 2N f ) c 2 E l) y z ε peak log (2N f ) b) Notched component The Similitude Concept states that if the local notch-tip strain history in the notch tip and the strain history in the test specimen are the same, then the fatigue response in the notch tip region and in the specimen will also be the same and can be described by the material strain-life (ε-n) curve. 28 Grzegorz Glinka. All rights reserved. 52

53 Information path for fatigue life estimation based on the da/dn-δk method F LOADING t GEOMETRY, K f PSO MATERIAL E ε Stress-Strain Analysis MATERIAL Damage Analysis da dn n ΔK th ΔK Fatigue Life 28 Grzegorz Glinka. All rights reserved. 53

54 Steps in the Fatigue Life Prediction Procedure Based on the da/dn-δk Approach a) Structure e) Stress, S ΔS 3 ΔS 4 b) Component ΔS 1 t Q H F ΔS 2 ΔS 5 c) Section with welded joint We ld R V P d) 28 Grzegorz Glinka. All rights reserved. 54

55 Steps in Fatigue Life Prediction Procedure Based on the da/dn-δk Approach (cont d) i) f) (x, y) Stress intensity factor, K (indirect method) Weight function, m(x,y) K = A K Y = πa n (, ) (, ) x y m x y dxdy a f Crack depth, a a i Fatigue Life Stress intensity factor, K (direct method) Number of cycles, N a g) K = 2πx I yfe FE or K = du E da = EG K Y = πa n h) Integration of Paris equation f ( ) i= 1 m Δ a = C ΔK ΔN i i i N a = a + Δa N = ΔN i i 28 Grzegorz Glinka. All rights reserved. 55

56 The Similitude Concept in the da/dn ΔK Method a) Structure 1-6 a b) Weld detail Q H F Crack Growth Rate, m/cycle a c) Specimen ΔK ΔK,MPa m P P The Similitude Concept states that if the stress intensity K for a crack in the actual component and in the test specimen are the same, then the fatigue crack growth response in the component and in the specimen will also be the same and can be described by the material fatigue crack growth curve da/dn - ΔK. 28 Grzegorz Glinka. All rights reserved. 56

57 What stress parameter is needed for the Fracture Mechanics based (da/dn( da/dn-δk) ) fatigue analysis? ρ S Stress (x) a T ( x) = K 2π x x The Stress Intensity Factor K characterizing the stress field in the crack tip region is needed! The K factor can be obtained from: - ready made Handbook solutions (easy to use but often inadequate for the analyzed problem) - from the (x) near crack tip stress or displacement data obtained from FE analysis of a cracked body (difficult) S - from the weight function by using the FE stress analysis data of un-cracked body (versatile and suitable for FCG analysis) 28 Grzegorz Glinka. All rights reserved. 57

58 3) Load/stress histories methods of assembling characteristic load/stress histories, the rainflow counting method live example, extraction of load/stress cycles from the frequency domain data. 28 Grzegorz Glinka. All rights reserved. 58

60 The load W and the nominal stress n in an railway axle 32M n,min =+ b 3 π d W a) 1 b) y W/2 W/2 B A W/2 x d 3 2 N.A. R B Moment M b l L Mc b 32Mb S = n = peak = = ± ; 3 I π d 3 W d π d Mb = ( L+ l) ; c = ; I = ; R A Note! In the case of smooth components, such as the railway axle, the nominal stress and the local peak stress are the same! 28 Grzegorz Glinka. All rights reserved. 6 32M n,min = b 3 π d n,max Stress n,min cycle c) n,a time

61 Loads and Stresses The load, the nominal stress, the local peak stress and the stress ss concentration factor F Axial load linear elastic analysis Stress peak n n = F A net ; ρ d n T y k n F = k F; = F 1 A net ; peak = hf; F Analytical, FEM Hndbk peak hf = ; hf = kf Kt; F F = k F; ni, F i peak, i = h F F ; i 28 Grzegorz Glinka. All rights reserved. 61

62 Loads and Stresses The load, the nominal stress, the local peak stress and the stress ss concentration factor M Bending load linear elastic analysis n Stress peak n = M c I net net ; ρ d n y T n = k M M; c k net M = ; = k ; ni, MM I i net b) M = h M or = k M K peak, i M i peak, i M i b t 28 Grzegorz Glinka. All rights reserved. 62

63 Stress Concentration Factors in Fatigue Analysis The nominal stress and the stress concentration factor in simple load/geometry configurations Tension S S Bending M Simple axial load P n = or S = A net Pure bending load P A gross r Stress peak n y n r Stress peak y net n = or S = Inet K t M c peak = or K t = S n M c I gross gross peak d n T S S 28 Grzegorz Glinka. All rights reserved. 63 d n T M net K t K t stress concentration factor (net or gross, net K t gross K t!! ) peak stress at the notch tip n - net nominal stress S - gross nominal stress gross K t

64 Stress concentration factors for notched machine components 3. Stress concentration factor K t = peak / n P h d b peak P n =P/[(b-d)h] Diameter-to-width ratio d/b (B.J. Hamrock et. al., ref.(26) 28 Grzegorz Glinka. All rights reserved. 64

65 Stress concentration factors for notched machine components 3. r Stress concentration factor K t = peak / n M peak Radius-to-height ratio r/h (B.J. Hamrock et. al., ref.(26) 28 Grzegorz Glinka. All rights reserved. 65 H h b Mc 6M n = = 2 I bh M H/h=6 H/h=2 H/h=1.2 H/h=1.5 H/h=1.1

66 Loads and Stresses The load, the nominal stress, the local peak stress and the stress ss concentration factor F M Simultaneous axial and bending load ni, = kffi+ kmmi; n ρ d n Stress peak T b) M F y or k K peak, i peak, i F, h t t, F, k K h M ; b t ; M ; = k F K + k F F i i t t = h F + h M M M i ; M i K b t From simple analytical stress analysis From stress concentration handbooks From detail FEM analysis ; 28 Grzegorz Glinka. All rights reserved. 66

67 Fluctuations and complexity of the stress state at the notch tip x 3 x 3 T 33 F F 22 x 2 22 x 2 T 2R 33 ρ t Non-proportional loading path Proportional loading path 23 A B C 23 A B C t 23 A t 23 A B 22 B 22 t C t C 28 Grzegorz Glinka. All rights reserved. 67

68 How to establish the nominal stress history? a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, M b =1, T=1 in order to establish the proportionality factors, k P, k M, and k T such that: k P k M τ k T = ; = ; = P M b T n P n M n T b) The peak and valleys of the nominal stress history n,,i are determined by scaling the peak and valleys load history P i, M b,i and T i by appropriate proportionality factors k P, k M, and k T such that: k P k M τ k T = = ; = P i M bi, T P M T ni, ni, ni, i c) In the case of proportional loading the normal peak and valley stresses can be added and the resultant nominal normal stress history can be established. Because all load modes in proportional loading have the same number of simultaneous reversals the resultant history has also the same number of resultant reversals as any of the single mode stress history. + = k P k M n, i P i M d) In the case of non-proportional loading the normal stress histories (and separately the shear stresses) have to be added as time dependent processes. Because each individual stress history has different number of reversals the number of reversals in the resultant stress history can be established after the final superposition of all histories. ; bi, n + i P i M b ( t ) = k P( t ) k M ( t ) i 28 Grzegorz Glinka. All rights reserved. 68

69 Superposition of nominal stress histories induced by two independent loading modes Two proportional modes of loading Two non-proportional modes of loading Stress n,a Mode a Stress n,a Mode a time Stress n,b Mode b Resultant stress: n = n,a + n,b Stress n Stress n,b Mode b time Resultant stress: n (t i )= n,a (t i )+ n,b (t i ) Stress n 28 Grzegorz Glinka. All rights reserved. 69

70 How to establish the linear elastic peak stress, peak, history? a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, M b =1, T=1 in order to establish the proportionality factors, k P, k M, and k T such that: = k P ; = k M τ = k T ; peak P peak M b peak T b) The peak and valleys of the notch tip peak stress history peak,,i are determined by scaling the peak and valleys load history P i, M b,i and T i by appropriate proportionality factors k P, k M, and k T such that: = k P = k M τ = k T ; peak, i P i peak, i M bi peak, i T, i c) In the case of proportional loading the normal peak and valley stresses can be added and the resultant notch tip normal peak stress history can be established. Because all load modes in proportional loading have the same number of simultaneous reversals the resultant history has also the same number of resultant reversals as any of the component single mode stress history. + = k P k M ; peak, i P i M b, i d) In the case of non-proportional loading the normal stress histories (and separately the shear stresses) have to be added as time dependent processes. Because each individual stress history has different number of reversals the number of reversals in the resultant stress history can be established after the final superposition of all histories. peak t i P t i + M b t i ( ) = k P ( ) k M ( ) 28 Grzegorz Glinka. All rights reserved. 7

71 Superposition of linear elastic notch tip stress histories induced by two independent loading modes Two proportional modes of loading Two non-proportional modes of loading Stress peak,a Stress peak,b Mode a Mode b Stress peak,b Stress peak,a Mode a time Mode b time Resultant stress: n = n,a + n,b Stress peak Stress peak Resultant stress: peak (t i ) = n,a (t i )+ n,b (t i ) 28 Grzegorz Glinka. All rights reserved. 71

72 Loading and stress histories Load history, F Nominal stress history, n Notch tip stress history, peak Load F F i F max Stress n n,i n,max Stress peak peak,i peak,max F i-1 F i+1 Time n,i-1 n,i+1 Time peak,i+1 Time peak,i-1 +1 ni, -1 i i-1 i+1 28 Grzegorz Glinka. All rights reserved. 72 Characteristic non-dimensional load/stress history ni, peak, i n, i = kf F F F i = = i = K max t ni, t F i ni, = K k F ni, n,max = peak, i peak,max

73 Wind load and stress fluctuations in a wind turbine blade Source [43] Wind speed [m/s] Load-lag stress [MPa] In-plane bending Time [s] Wind speed [m/s] Load-lag stress [MPa] Out of plane bending Time [s] Note! One reversal of the wind speed results in several stress reversals Wind speed fluctuations + Blade vibrations Stress fluctuations Grzegorz Glinka. All rights reserved. 73

74 Characteristic load/stress history in the aircraft wing skin a) b) c) Stress Flying Taxiing Landing time Source [9] a) Ground loads on the wings, b) Distribution of the wing bending moment induced by the ground load, c) Stress in the lower wing skin induced by the ground and flight loads 28 Grzegorz Glinka. All rights reserved. 74

75 Typical load fluctuations in various engineering objects max peak F F max peak max peak Stress at rear motor-car axle Pressure in a reactor +1 max peak Stress at motor-car wheel max peak F F max peak max peak M M max peak -1 Torsion moment at steel-mil shaft M M max peak M M max peak Bending moment at stub axle of motor-car G G max peak p p max peak Vertical acceleration of the Gravity Center of fighter airplane Pressure in an oil-pipeline G G V max Vpeak Vertical acceleration of the Gravity Center of transport airplane 28 Grzegorz Glinka. All rights reserved. 75

76 How to get the nominal stress n from the Finite Element method stress data? Notched shaft under axial, bending and torsion load x 3 M F d x 2 T F M T D 33 r t Discrete cross section stress distribution obtained from the FE analysis x 3 n peak a) Run each load case separately for an unit load b) Linearize the FE stress field for each load case d Grzegorz Glinka. All rights reserved. 76

77 How to establish the link between the fluctuating load history and the stress distribution, (x,y), in the potential crack plane? a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, M b =1, T=1 in order to establish the link between the load and the stress distribution, k P, k M, and k T such that: = (, ) P; = k ( x, y) M τ ) = k ( x, y) T ( x, y) k x y ( x, y) ; ( x, y P M b T Where: (x,y) stress distribution in the x-y plane (crack plane) k P (x,y), k M (x,y), k T (x,y) reference stress distributions induced by unit loads P=1, M b =1, T=1 b) The fluctuating stress distributions corresponding to instantaneous peaks and valleys of the load history are determined by scaling the reference stress distributions k P (x,y), k M (x,y), k T (x,y) by appropriate magnitudes of the load history P i, M b,i and T i such that: x y = x y =, ) τ = ( x, y) T (, ) k ( x, y) P; (, ) k ( x y M ; ( x, y) k i P i i M bi, i T c) In the case of proportional loading the stress distributions corresponding to peaks and valleys of the load history can be added and the resultant stress distributions can be established. The nominal stress history can be also used as the stress distribution calibration parameter. + ( x, y) = k ( x, y) P k ( x, y) M i ; i P M bi, 28 Grzegorz Glinka. All rights reserved. 77 y = k ( x, y) k ( x, y) i np +, nm, ( x, ) np, i nm, i ; i

78 How to get the resultant stress distribution from the Finite Element stress data? (Notched shaft under axial, bending load) x 3 n b x 3 d Bending 22 P M d x 2 M P D 33 t r x 3 n m x 3 peak d Axial 22 d Resultant (x 3 ) Grzegorz Glinka. All rights reserved. 78

79 a) Determination of approximate stress distributions in notched bodies (for simultaneous axial and bending load) S M b) Note! Good accuracy for x < 3.5ρ r d n Stress peak yy n xx T x n r Stress d n peak yy xx κ T x or yy yy = peak x 1 x 1 1 x + 2 r r κ 2 4 n r r κ Kt = x + x + x 2 κ (for pure axial load)! κ = distance to the neutral axis (for pure bending load)! peak 1 1 x x x 1 1 x 1 1 x xx = r 2 42 r 2 6 r 2 r κ peak K t = or n K t n 1 1 x 1 1 x 1 1 x 1 1 x xx = r 2 42 r 2 6 r 2 r x κ 28 Grzegorz Glinka. All rights reserved. 79

80 Cyclic nominal stress and corresponding fluctuating stress distribution ibution n, max Stress n n, time n, min x 3 22 (x 3, n,min ) 22 (x 3, n, ) 22 (x 3, n,max ) Resultant d Grzegorz Glinka. All rights reserved. 8

history a3799_1 with removed ranges less than 5% of the

83 The complete Right Hand Axle Torque stress/load history Stress history a3799_1 (Right Hand Axle Torque) as recorded Stress history a3799_1 with removed ranges less than 5% of the largest one in the history 28 Grzegorz Glinka. All rights reserved. 83

84 Details of the stress history a3799_c1; visible repeatable working cycles; (Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the history) S max,1 S max,1 = S max,2 S max,4 Smax,3 S max,i 28 Grzegorz Glinka. All rights reserved. 84

85 Distribution of peak stresses S max in the stress history a3799_1; (Right Hand Axle Torque History) ReliaSoft Weibull Probability - Lognormal Probability-Lognormal Max_1 Lognormal-2P RRX SRM MED FM F=87/S= Data Points Probability Line m = MPa μ = 7.21 MPa COV=.198 Unreliability, F(t) Grzegorz Glinka University of Waterloo 22/5/27 7:24:1 AM m=4.1795, s=.195, r=.9831 (a3799_1) Stress, (MPa) 28 Grzegorz Glinka. All rights reserved. 85

Non-dimensional stress history a3799_1; (Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the history) Statistical data for the scaling peak stress S max

86 Non-dimensional stress history a3799_1; (Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the history) Statistical data for the scaling peak stress S max Log-Normal Probability distribution, LN (4.1975,.198) Mean Value Standard Deviation Coefficient of variation m = MPa μ = 7.21 MPa COV=.198 (actual) m = 1 μ =.198 COV=.198 (Non-dimensional) 28 Grzegorz Glinka. All rights reserved. 86

89 Stress/Load Analysis - Cycle Counting Procedure and Presentation of Results The measured stress, strain, or load history is given usually in the form of a time series, i.e. a sequence of discrete values of the quantity measured in equal time intervals. When plotted in the stress-time space the discrete point values can be connected resulting in a continuously changing signal. However, the time effect on the fatigue performance of metals (except aggressive environments) is negligible in most cases. Therefore the excursions of the signal, represented by amplitudes or ranges, are the most important quantities in fatigue analyses. Subsequently, the knowledge of the reversal point values, denoted with large diamond symbols in the next Figure, is sufficient for fatigue life calculations. For that reason the intermediate values between subsequent reversal points can be deleted before any further analysis of the loading/stress signal is carried out. An example of a signal represented by the reversal points only is shown in slide no The fatigue damage analysis requires decomposing the signal into elementary events called cycles. Definition of a loading/stress cycle is easy and unique in the case of a constant amplitude signal as that one shown in the figures. A stress/loading cycle, as marked with the thick line, is defined as an excursion starting at one point and ending at the next subsequent point having exactly the same magnitude and the same sign of the second derivative. The maximum, minimum, amplitude or range and mean stress values characterise the cycle. 1 cycle Δ = max min stress range max min a = stress amplitude 2 max + min m = mean stress 2 max min Unfortunately, the cycle definition is not simple in the case of a variable amplitude signal. The only nondubious quantity, which can easily be defined, is a reversal, example of which is marked with the thick line in the Figures below. Stress m a time 28 Grzegorz Glinka. All rights reserved. 89

92 Constant and Variable Amplitude Stress Histories; Definition of the Stress Cycle & Stress Reversal a) Constant amplitude stress history max Stress One cycle mean min Δ = a max min Δ max = = 2 2 ; min b) Variable amplitude stress history Time m = max + 2 min ; Stress One reversal Time R = min max 28 Grzegorz Glinka. All rights reserved. 92

93 Stress Reversals and Stress Cycles in a Variable Amplitude Stress History The reversal is simply an excursion between two-consecutive reversal points, i.e. an excursion between subsequent peak and valley or valley and peak. In recent years the rainflow cycle counting method has been accepted world-wide as the most appropriate for extracting stress/load cycles for fatigue analyses. The rainflow cycle is defined as a stress excursion, which when applied to a deformable material, will generate a closed stress-strain strain hysteresis loop. It is believed that the surface area of the stress-strain hysteresis loop represents the amount of damage induced by given cycle. An example of a short stress history and its rainflow counted cycles content is shown in the following Figure. 28 Grzegorz Glinka. All rights reserved. 93

94 Stress History and the Rainflow Counted Cycles Stress history Rainflow counted cycles Stress i-1 i+1 i-2 i+2 Time i A rainflow counted cycle is identified when any two adjacent reversals in the stress history satisfy the following relation: ABS ABS i 1 i i i Grzegorz Glinka. All rights reserved. 94

95 The Mathematics of the Cycle Rainflow Counting Method for Fatigue Analysis of Stress/Load Histories A rainflow counted cycle is identified when any two adjacent reversals in thee stress history satisfy the following relation: ABS The stress amplitude of such a cycle is: a = ABS < ABS i 1 2 i 1 i i i+ 1 i The mean stress of such a cycle is: i 1 + m = 2 The stress range of such a cycle is: Δ = ABS i i 1 i 28 Grzegorz Glinka. All rights reserved. 95

96 The ASTM rainflow cycle counting procedure - example Determine stress ranges, ΔS i, and corresponding mean stresses, S mi for the stress history given below. Use the ASTM rainflow counting procedure. S i =, 4, 1, 3, 2, 6, -2, 5, 1, 4, 2, 3, -3, 1, -2 (units: MPa 1 2 ) 6 5 Stress S i (MPa 1 2 ) Reversing point number, i 28 Grzegorz Glinka. All rights reserved. 96

97 1. Find the absolute maximum revesal point; Rainflow cycle counting The Direct Method: 2. Add the absolute maximum at the end of the history, i.e. make it to be the last reversing point in the history; 3. Start counting form the reversal no. 3 (always); Stress History Absolute maximum Stress (MPa)x Stress (MPa)x Starting point Reversal point No. 28 Grzegorz Glinka. All rights reserved. 97

101 Stress (MPa)x Starting point Δ = = = = = = ( 3) 9; 6 3 m, ; Cycles counted Direct method =1; m,3-4 = 2.5; =3; m,1-2 = 2.5; =1; m,1-11 = 2.5; =3; m,8-9 = 2.5; =7; m,6-7 = 1.5; =3; m,13-14 =-.5; =9; m,5-12 = 1.5; 28 Grzegorz Glinka. All rights reserved. 11

104 The stress ranges are often grouped into classes and the final result of the rainflow counting is presented in the form of an exceedance diagram or a cumulative frequency distribution. The character of the exceedance diagram depends on the loading conditions and the dynamics property of the system within which the given component is working. It is generally known that the character of the frequency distribution diagram does not depend strongly on the loading conditions and it remains almost the same for a given system or machine. Therefore, some kind of standard frequency distribution (exceedance) diagrams can be found for similar types of machines and structures such as cranes, aeroplanes, offshore platforms, etc. The frequency distribution diagram can also be interpreted as a probability density distribution when presented in n j /N T vs. Δ j /Δ max co-ordinates. The experimental probability density, f(x j ), for stress range, Δ j, is determined as: ( ) j f xj = NT The discrete experimental probability density function, represented by the rectangular bars can be approximated by a continuous function. The attempt is usually made to fit one of the well-known theoretical probability density distributions such as the Normal, Beta, Log-Normal, Weibull, or others. The probability density data can also be converted into the sample cumulative probability diagram according to the relation: 28 Grzegorz Glinka. All rights reserved. 14 n i= j ( < x ) = j F x i= 1 n N i T

106 a) The stress range exceedance diagram (stress spectrum) Stress range Δ j Total number of cycles in the entire history, N T Δ max n j=6 - number of cycles Δ j=6 in the class j=6 1 Δ max /Δ max Relative stress range Δ j / Δ max Δ j=4 /Δ max j=4 n j=6 /N T j=4 j=6 j=7 Number cycles N j=6 j= b) the stress range frequency distribution diagram Relative number of cycles N/N T 28 Grzegorz Glinka. All rights reserved. 16

107 a) Probability Diagram Obtained from the Experimental Cumulative Spectrum; b) Cumulative Probability F(x<x j ) Probability f(x j )=n j /N T a) Cumulative prob., F(x<x j )=Σn j /N T j=7 j=6 j=5 j=4 j=3 j=2 j=1 j=7 j=6 j=5 j=4 j=3 j=2 j= Relative stress range, x j = Δ j /Δ max b) n + n + n n 4 N T n + n + n n + 4 n 3 N T n + n + n + n n + 3 n 2 N T.5 1. Relative stress range, x j = Δ j /Δ max Continuous probability distribution function: p(x), p( / max ), p(s a /S a,max ) 28 Grzegorz Glinka. All rights reserved. 17

108 a) Probability Density Diagram obtained from the Exper. Cumulative Spectrum; b) Cumulative Probability F(x<x j ) ( ) f xj nj NT p ( x j ) = = ; Δx Δx ( j < j j) ( j) Δ xj = nj NT; P x x < x +Δ x = p x Probability density, p(x j ) p(x 7 )=f(x 7 )/ x 7 =n 7 /N T / x 7 j j Δx 6 Δx 5 Cumulative prob., P(x<x j )=Σn j /N T Grzegorz Glinka. All rights reserved. 18 j ( ) ( ) j=7 j=6 j=5 j=4 j=3 j=2 j=1 P x < x = p x Δ x = n N.5 1. Relative stress range, x j = Δ j /Δ max j j j j T 1 1 n + n + n n 4 N T n + n + n n + 4 n 3 N T n + n + n + n n + 3 n 2 N T j=7 j=6 j=5 j=4 j=3 j=2 j=1 j.5 1. Relative stress range, x j = Δ j /Δ max Continuous probability density distribution: p(x), p( / max ), p(s a /S a,max )

109 The use of the probability density distribution Given are: The maximum stress amplitude, S a,max and the total number of cycles in the spectrum N T and the probability density function p(s a /S a,max )= p(x); The task of the analyst is to determine the number of applications (cycles) n j of stress amplitudes within the interval (S a,j < S a < S a,j+1 ); Find how many cycles n j with amplitudes S a,j < S a < S a,j+1 are to be found out in the total population of N T load cycles to be applied. x j =S a,j /S a,max and x j+1 =S a,j+1 /S a,max. Number of cycles in the class j: x j+1 n = N p(x)dx j T x Average stress in the class j: j S a,j = S S 2 + a,j a,j Grzegorz Glinka. All rights reserved. 19

110 Various Loading Spectra a) constant amplitude, b) mixed spectrum, c) Gaussian normal spectrum, d- e) mixed spectra Relative stress amplitude, a,i / max a 1. b.2 c.2 d.2 e.2 f (t) (t) (t) a d f t t t Relative number of cycles, N i /N Total 28 Grzegorz Glinka. All rights reserved. 11

111 Ways of representing the stress signal Stress signal Time Domain Representation Stress MPa Sequence of peaks and valleys Stress signal Frequency Domain Representation Stress amp. MPa Amplitudes associated with certain frequencies 28 Grzegorz Glinka. All rights reserved. 111 Courtesy of D. Socie, Univ. of Illinois

112 A Frequency Domain approach to fatigue analysis of random cyclic stress processes In the case of vibration induced stresses and strains the determination of the stress- or strain-time history as discussed earlier is rather difficult and very often the stress signal/process can not be determined in a deterministic form. Therefore concepts taken from random process theory, such as the power spectral density (PSD), root mean square (RMS) and the Fourier transform analysis are used to characterize random load and stress histories. The task of the analyst is determine the most probable stress spectrum or the probability density distribution of stress amplitudes of the analysed object knowing the statistical properties (PSD, RMS, etc.) of the stress process at the critical location. The stress spectrum determined in the form of discrete classes of stress amplitudes or in the form of a probability density distribution is subsequently used for the determination of cumulative fatigue damage and fatigue durability. Therefore some elements of the random process theory need to be reviewed including the physical interpretation of basic parameters used for the characterisation of a random process such as the stress fluctuations induced by a random excitation signal. 28 Grzegorz Glinka. All rights reserved. 112

113 General fatigue analysis procedures Load time history Structural model Stress time history ( TIME DOMAIN ) Stress Cycles Miner fatigue damage FATIGUE LIFE ( FREQUENCY DOMAIN ) Acceleration PSD Transfer function Stress PSD 28 Grzegorz Glinka. All rights reserved. 113

The Time and Frequency Domain analysis of random cyclic stress signal RANDOM PROCESS

115 What s s a PSD? The technique was pioneered by electronics engineers in the 194s. They used it to characterise noise in electronic circuits. They were interested in the average amplitude of noise at different frequencies but couldn t calculate the Fourier Transform at that time Increasing filter frequency 5 5 frequency 5 Extract time signal Low pass filter over a range of frequencies frequency 5 frequency PSD = change of mean 2 vs. frequency Plot cumulative mean 2 vs. filter frequency Calculate mean value (i.e. mean square amplitude) Square filtered time signal to remove negatives 28 Grzegorz Glinka. All rights reserved. 115

116 X(t) Descriptors of random processes Root Mean Square (RMS) t RMS + 2 = [ x] p(x)dx T 1 T T ( ) X t 2 dt X i or t RMS N T i= 1 = N x T 2 i N T The RMS is equal to the standard deviation x when the mean value is zero, i.e. when μ x =! 28 Grzegorz Glinka. All rights reserved. 116

117 X(t) x max x min X(t) x max Descriptors of random processes ( continued) The meaning and magnitude of the RMS parameter t RMS μ x Δx RMS=.77x max μ x =.637x max Δx= x max -x min =2x max = 2x a Crest factor =x max /RMS=1.414 RMS=x max μ x =x max t Δx= x max -x min =2x max = 2x a X(t) x max x min x max Crest factor =x max /RMS=1 RMS=.577x max t μ x =.5x max Δx= x max -x min =2x max = 2x a x min 28 Grzegorz Glinka. All rights reserved. 117 Crest factor =x max /RMS=1.733

The Nature of Frequency Analysis the light spectrum analogy Fourier Transform is the

The Fourier Transform After replacing the n/t term by frequency f the Fourier transform equations can be written as: Fourier Transform from time to frequency domain: Inverse Fourier Transform from

121 The Fourier Transform After replacing the n/t term by frequency f the Fourier transform equations can be written as: Fourier Transform from time to frequency domain: Inverse Fourier Transform from frequency to time domain: + i 2 f t X f X t e dt ( ) 2 ( ) = T X( f ) = n X( tk) e N + i 2π f t X t X f e df π ( ) 2 ( ) 2π n i k N 2 T T k Continuous process Discrete process Inverse Fourier Transform = ( ) ( ) X t = k X tk e T 2π n i k N 1 T n Time domain Frequency domain Fourier Transform 28 Grzegorz Glinka. All rights reserved. 121

122 The Number of Cycles Stress + P t Upward zero crossing, + E[ + ] Expected number of upward zero crossing + Peak P E[P] Expected number of peaks 1 sec IF ( + ) E 3 = = EP 5 ( ) Irregularity factor G(f), (Stress 2 )/Hz G k (f) The n th moment area under PSD: Number of upward n ( ) m = f G f δf zero crossing: ( ) n E + = m m 2 f k SO Rice, 1954 δf Frequency f, [Hz] E P m Number of peaks: ( ) 4 = m 2 28 Grzegorz Glinka. All rights reserved. 122

The Cycle Ranges narrow band process ΔS p( Δ S) = e 4m 2 ΔS 8m N( S) S ( S) 2 JS

University of Waterloo. Partial notes Part 6 (Welded Joints) Fall 2005

University of Waterloo. Partial notes Part 6 (Welded Joints) Fall 2005 University of Waterloo Department of Mechanical Engineering ME 3 - Mechanical Design 1 artial notes art 6 (Welded Joints) (G. Glinka) Fall 005 1. Introduction to the Static Strength Analysis of Welded