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© 2008 Grzegorz Glinka. All rights reserved. 1

Fatigue Design of Welded Structures Fatigue Design of Welded Structures --

Physical Physical Modeling vs. Empirical RulesModeling vs. Empirical Rules

G. G. GlinkaGlinka, Ph.D., D.Sc., Ph.D., D.Sc.

University of Waterloo,Department of Mechanical and Mechatronics

Engineering,

200 University Ave. W, Waterloo,

Ontario, Canada N2L 3G1

Phone: 1-519-888 4567 ext. 33339

e-mail: [email protected]

Stockholm, 21-22, October 2008

mailto:[email protected]


ContentsContents1) 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,


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;


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


1.

Bannantine, J., Corner, Handrock, Fundamentals of Metal Fatigue Analysis, Prentice-Hall, 1990.... (complete, correct, up to date)

2. Dowling, N., Mechanical Behaviour of Materials, Prentice Hall, 1999...

( 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, 2006 (understandable, rigorous, mechanics perspective)

4. Stephens, R.I., Fatemi, A., Stephens, A.A., Fuchs, H.O., Metal Fatigue in Engineering, John Wiley, 2001.... (good general reference, well done)

5. Radaj, D., Design and Analysis of Fatigue Resistant Structures,

Halsted

Press, 1990..... (Complete, civil engineering analysis perspective)

6.Suresh, S., Fatigue of Materials,

Cambridge University Press, 1998.... (Encyclopedic, up to date, fundamental metallurgical perspective)

7. Anderson, T.L., Fracture Mechanics-Fundamentals and Applications, CRC Press, Boca Raton, 1995

8. Collins, J. A., Failure of Materials in Mechanical Design, John Wiley & Sons, New York, 1993

9. Broek, D., Elementary Engineering Fracture Mechanics, Martinus

Nijhoff, The Hague, 1982

10. Hertzberg, R.W., Deformation and Fracture Mechanics of Engineering

Materials, John Wiley, New York, 1989.

BibliographyBibliography


11. K. Iida and T. Uemura, “Stress Concentration Factor Formulas Widely Used in Japan”, Document IIW XIII-1530-94, The International Welding Institute, 1994.

12. Bahram

Farhmand, G. Bockrath

and J. Glassco, Fatigue and Fracture Mechanics of High Risk Parts, Chapman @Hall, 1997

13. Sandor, R.J., Principles of Fracture Mechanics,

Prentice Hall, Upper Saddle River, 2002.

14. Socie, D.F., and Marquis, G.B., Multiaxial

Fatigue, Society of Automotive Engineers, Inc., Warrendale, PA, 2000.

15. 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, 1980 (in French).

19. Haibach, E., Betriebsfestigkeit, VDI Verlag, Dusseldorf, 1989 (in German).

20. Barsom, J.M, and Rolfe, S.T., Fracture and Fatigue Control in Structures, Prentice Hall, New Jersey, 1987.

21. Courtney, T.H., Mechanical Behavior

of Materials, McGraw Hill, New York, 1990.

Bibliography Bibliography (continued)(continued)


22. Collins, J.C., Mechanical Design of Machine Elements and Machines, John Wiley & Sons, New York, 2003.

23. Shigley, J.E., and Mischke, C.R., Mechanical Engineering Design, McGraw-Hill, New York, 2001

24. Juvinal, R.C., and Marshek, K.M., Fundamentals of Machine Components Design, John Wiley & Sons, New York, 2000.

25. Norton, R.L., Machine Design –

An Integrated Approach, Prentice Hall, New Jersey, 2000

26. Hamrock, B.J., Jacobson, Bo and Schmid, S.R., Fundamentals of Machine Elements, McGraw-

Hill, Boston, 1999.

27. Orthwein, W., Machine Component Design, West Publishing Company, New York, 1990.

28. 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, pp723-729.

29. V.A. Ryakhin

and G.N. Moshkarev, Durability and Stability of Welded Structures in Earth Moving Machinery”, Mashinostroenie, Moscow, 1984 (in Russian)

30. 7. V. I. Trufyakov

(editor), The Strength of Welded Joints under Cyclic Loading, Naukova

Dumka, Kiev, 1990 (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. 223-241.



32. S.V. Petinov, Principles of Engineering Fatigue Analyses of Ship Structures, Sudostroyenie, Leningrad, 1990 (in Russian and in English)

33. C.C. Monahan, Early Fatigue Cracks Growth at Welds, Computational Mechanics Publications, Southampton UK, 1995.

34. Y. Murakami et. al, (editor), Stress Intensity Factors Handbook,

Pergamon

Press, Oxford, 1987

35. O.W. Blodgett, Design of Welded Structures, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1966

36. A. Almar-Naess, Fatigue Handbook –

Offshore Steel Structures, Tapir Publishers, Trondheim, Norway, 1985

37. E. Niemi, Structural Stress Approach To Fatigue Analysis Of Welded Components,

The International Institute of Welding, Doc. XIII-WG3-06-99, 1999

38. D. Pingsha, A Structural Stress Definition and Numerical Implementation for Fatigue Analysis of Welded Joints, International Journal of Fatigue, vol. 23, 2001, p. 865-876

39. R.E. Little andJ.C. Ekvall, Statistical Analysis of Fatigue Data, ASTM STP 744, 1981

40. 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, 1978

41. J.G. Hicks, Welded Joint Design, Abington Publishing, UK, 1997-edition 2 and 1999-edition 3.



Technical Report on Fatigue Properties

-

SAE J1099, SAE Book of Standards, 1992, SAE, p. 3.76

Standard Practices for Cycle Counting in Fatigue Analysis, ASTM Standard E 1049 -

85, ASTM Book of Standards, vol. 03.01, 1994, p. 789

Standard Recommended Practice for Contant-Amplitude Low-Cycle Fatigue Testing, ASTM Standard E 606 -


Standard Practice for Stastical

Analysis of Linearized

Stress-Life (S-N) and Strain-Life (ε-N) Fatigue Data, ASTM Standard E 739 -


Standard Test Method for Measurement of Fatigue Crack Growth Rates, ASTM Standard E 647 -

91, ASTM Book of Standards, vol. 03.01, 1996.

Relevant StandardsRelevant Standards


Relevant Relevant wwebsitesebsites

--

FatigueFatiguehttp://www.tech.plym.ac.uk/sme/Interactive_Resources/tutorials/FractureMechanics/index.html

(**Good basic course notes and tutorials on Fracture Mechanics)

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/anal/kelly/fatigue.html#life

(*good overview notes)

http://www.engr.sjsu.edu/WofMatE/FailureAnaly.htm

(*Failure of Materials, examples)

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/exper/ballard/www/ballard.html

(*General notes on fatigue)

http://highered.mcgraw-hill.com/sites/0072520361/student_view0/machine_design_tutorials.html

(*Tutorials on Strength theories, Stress concentration factors and Pressure vessel design)

http://emat.eng.hmc.edu/Calculators/shaft/shaft_design.htm

(*Calculator for shaft design, static and fatigue)

http://pergatory.mit.edu/2.75/software_tools/software_tools.html

(*Excel Files, stress concentration factors for shafts and design of shafts)

http://www.roymech.co.uk/Useful_Tables/Drive/Shaft_design.html

(*Interesting notes on shaft design and Tables)

http://www.fatiguecalculator.com/

(Stress concentration factors, fatigue methods)

http://www.saffd.com/

(*Fatigue software, manual and theory of fatigue methods)

http://www.tech.plym.ac.uk/sme/Interactive_Resources/tutorials/FractureMechanics/index.html

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/anal/kelly/fatigue.html#life

http://www.engr.sjsu.edu/WofMatE/FailureAnaly.htm

http://www.sv.vt.edu/classes/MSE2094_NoteBook/97ClassProj/exper/ballard/www/ballard.html

http://highered.mcgraw-hill.com/sites/0072520361/student_view0/machine_design_tutorials.html

http://emat.eng.hmc.edu/Calculators/shaft/shaft_design.htm

http://pergatory.mit.edu/2.75/software_tools/software_tools.html

http://www.roymech.co.uk/Useful_Tables/Drive/Shaft_design.html




Relevant Websites Relevant Websites ––

Fracture MechanicsFracture Mechanicshttp://www.tech.plym.ac.uk/sme/Interactive_Resources/tutorials/FractureMechanics/index.html

(**Good basic course notes and tutorials on Fracture Mechanics)

http://simscience.org/cracks/advanced/cracks1.html

(*Cracking of dams, history of FM, animations)

http://www.mech.uwa.edu.au/DANotes/fracture/LEFM/LEFM.html

(**Good notes on FM from Australia)

http://www.modares.ac.ir/eng/mmirzaei/

(**Good complete notes on Fracture Mechanics)

http://www.efunda.com/formulae/solid_mechanics/fracture_mechanics/fm_intro.cfm

(**Concise notes/explanations of basics of FM)

http://en.wikipedia.org/wiki/Fracture_mechanics

(* Fracture Mechanics –

the Wikipedia

way)

http://www.engin.brown.edu/courses/En222/Notes/Fracturemechs/Fracturemechs.htm

(**Plastic, brittle, fatigue fractures –

notes)

http://wwwold.ccc.commnet.edu/lta/lta4/index.pdf#search=%22Calculation%20of%20Stress%20Intensity%2

0Factors%22

(**NASA practical use of Fracture Mechanics –

good examples)

http://mysopromat.ru/uchebnye_kursy/mehanika_razrusheniya/spravochnye_dannye/funktsii_dlya_raschet

a_kin/

(Catalogue of SIF)

http://www.tech.plym.ac.uk/sme/Interactive_Resources/tutorials/FractureMechanics/index.html

http://simscience.org/cracks/advanced/cracks1.html

http://www.mech.uwa.edu.au/DANotes/fracture/LEFM/LEFM.html

http://www.modares.ac.ir/eng/mmirzaei/

http://www.efunda.com/formulae/solid_mechanics/fracture_mechanics/fm_intro.cfm

http://en.wikipedia.org/wiki/Fracture_mechanics

http://www.engin.brown.edu/courses/En222/Notes/Fracturemechs/Fracturemechs.htm

http://wwwold.ccc.commnet.edu/lta/lta4/index.pdf#search=%22Calculation%20of%20Stress%20Intensity%20Factors%22

http://wwwold.ccc.commnet.edu/lta/lta4/index.pdf#search=%22Calculation%20of%20Stress%20Intensity%20Factors%22

http://mysopromat.ru/uchebnye_kursy/mehanika_razrusheniya/spravochnye_dannye/funktsii_dlya_rascheta_kin/

http://mysopromat.ru/uchebnye_kursy/mehanika_razrusheniya/spravochnye_dannye/funktsii_dlya_rascheta_kin/


1) Brief review of American and European 1) Brief review of American and European rules concerning static strength analysis rules concerning static strength analysis of of weldmentsweldments,,

•

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


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 τ1

and

τ2

and one normal stress component σ1

.


A Welded Structure –

Example

VP

Q

R

H

F

Weld

a) Structure

b) Component

c) Section with welded joint

Aσn

d) Weld detail A

Weld A

σσ

F


Fillet welds

Butt welds

Welded segment of an excavator Welded segment of an excavator armarm


P1

P2

M

Q

Internal loads and weld arrangmentInternal loads and weld arrangment

in a section of an in a section of an excavator armexcavator arm


Stress flowStress flow

in a in a welded welded section of an excavator armsection of an excavator arm

σ

τ

τ

τ

σ

τ

τ

τ

τ


g =

hθ

r

t

l = hp

PP MM

Typical geometrical weld configurations

TT--joint with fillet weldsjoint with fillet welds

Butt welded jointButt welded joint

t

r

t1

= tp

θ

hp

h

PP

y

x

MM


Stress concentration & stress distributions in weldments

Various stress distributions in a butt weldment;

r

A

B

C

DE

P

M

F

σpeak

σhs

σn

σpeak

t

C

σn

• 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).


Stress components in the weld throat cross Stress components in the weld throat cross section of a butt weldmentsection of a butt weldment

P

R

στ

L

t

R

P

σ

= P/A

τ

= R/A

Resultant equivalent stress

( )eq22 3σσ τ= +

A = t·L


Various stress distributions in a T-butt weldment with transverse fillet welds;

r

t

t1

ED

BC

A

σpeak

σn

σhs

FP

M

C

Θ

• 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).

Stress concentration & stress distributions in weldments


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.

NonNon--load load carrying weldscarrying welds

Load carrying Load carrying weldswelds


TT--

butt weldment with nonbutt weldment with non--

loadload--carrying transverse carrying transverse fillet welds fillet welds

(static strength analysis not required!!)(static strength analysis not required!!)

RV

P

L

t

R

V

P

h


Stress components in the weld throat cross section plane in a Stress components in the weld throat cross section plane in a TT--

butt weldment with loadbutt weldment with load--carrying transverse fillet weldscarrying transverse fillet welds(nominal stress components in the weld throat cross section!!)(nominal stress components in the weld throat cross section!!)

A = L·t

= 2L·h·sinα

τ1

στ2

RP

L

t

RP

σn

α

h

σ

= P·cosα/A; τ1 = P·sinα/A; τ2 = R/A



butt weldment with load butt weldment with load --

carrying transverse fillet weldscarrying transverse fillet welds

(simplified combination of stresses in the weld throat cross sec(simplified combination of stresses in the weld throat cross section according tion according

to the customary American method !!)to the customary American method !!)

σ

= 0 !!;

τ1

= σX

=P/A;

τ2

=R/A

Calculation of stresses in a fillet weld

τ1

σxτ2

RP

L

t

RP

σn

α

(It is assumed that only shear stresses act in the weld throat !)

A = L·t

= 2L·h·sinα



butt weldment with loadbutt weldment with load--carrying transverse fillet weldscarrying transverse fillet welds

(the customary American method !!)(the customary American method !!)

τ1

τ2

RP

L

t

RP

σn

α

τ

( ) ( )2 21 2ττ τ= +

( ) ( )2 21 2

3

3

eqσ τ

τ τ

=

⎡ ⎤= +⎣ ⎦

Equivalent stressThe resultant shear stress

Calculation of stresses in a fillet weld


b)

c)

Fillet welds under primary shear and bending loadFillet welds under primary shear and bending load

2

1,

2

2M

MM

dV l V lt b dt b d

M cIσ

τ σ

⋅ ⋅ ⋅= = ⋅ ⋅⋅ ⋅⋅=

=

σV

σV

σM

V

σM

d)

shear shear

σV

V

σV

1,

V

VV

Vt bσ

τ σ

=⋅

=

M

σM

σM

bending bending

ht

b

d

l

V

α

a)


Fillet welds in primary shear and bending:Fillet welds in primary shear and bending:the American customary method of combining the primary shear andthe American customary method of combining the primary shear and

bending stressesbending stresses

(according to (according to R.C.JuvinalR.C.Juvinal

& K.M. & K.M. MarshekMarshek

in Fundamentals of Machine Component Design, Wiley, 2000)in Fundamentals of Machine Component Design, Wiley, 2000)

))

2 2 2 21, 1,

222

2

1

1 1

V M V M

ld

V V l Vt b t b d t b

τ

τ

σ σ σ τ τ

⎛ ⎞⎛ ⎞⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟

⎝ ⎠ ⎝ ⎠

= + = +

= + = +

=

⋅⋅ ⋅ ⋅ ⋅

Acceptable design:

2

2

2

2 3 1

1 3

31

ys

ys or ld ys

ld

V Vt b t b σ

τ σ

σ ⎛ ⎞+⎜ ⎟

⎝≤

⎠≤

≤

+⋅ ⋅

d)

σV

σM

V

σ τ1

= σ

σV

σM

σ τ1

= σ


22

2M

dV l V lt b dt b d

M cIσ

⋅ ⋅ ⋅= = ⋅ ⋅⋅ ⋅⋅=V

Vt bσ = ⋅

a)

σV

V

σV

,V nσ ,V tτ

VVt bσ = ⋅

a)

σV

V

σV

,V nσ ,V tτ b)

M

σM

σM

σM.n

τM.t

b)

M

σM

σM

σM.n

τM.t

2 2

, , , ,

2 2 222

2 2 1

3

45

2

eq ysV n M n V t M t

eq ysV V M M

for

V l lt b d d

σ σ σ τ τ σ

α

σ σ σ σ σ σ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

⎛ ⎞⎜ ⎟

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

⎝ ⎠

= + +

− +⋅

− ≤

=

= − + = ≤

Fillet welds in primary shear and bending:Fillet welds in primary shear and bending:the ISO/IIW method of combining the primary shear and bending shthe ISO/IIW method of combining the primary shear and bending shear stressesear stresses


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 τ1

and τ2

. Therefore the resultant shear stress can be determined as:

2 21 2τ τ τ= +

The weld is acceptable if :

3ys

ys

στ τ< =

Where:

τys

is the shear yield strength of the: weld metal for fillet welds

and parent metal for butt welds

Static Strength Assessment of Fillet WeldsStatic Strength Assessment of Fillet Welds



butt weldment with loadbutt weldment with load--carrying transverse fillet weldscarrying transverse fillet welds(correct solid mechanics combination of stresses in the weld thr(correct solid mechanics combination of stresses in the weld throat oat ––

the the European approach)European approach)

τ1

στ2

RP

L

t

RP

σn

σ

= Pcosα/A

τ1 = Pcosα/A

τ2 = R/A

A = 2L·t·cosα

α

Resultant equivalent stress

( ) ( )2 221 23eqσ σ τ τ⎡ ⎤= + +⎣ ⎦


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 σ

perpendicular to the throat plane, the shear stress τ2

acting in the throat parallel to the axis of the weld and a shear stress component τ1

acting in the throat plane and being perpendicular to the longitudinal axis of the weld. The proposed formula for calculating the equivalent

stress is:

( )2 2 21 1 23eqσ β σ τ τ= + +

The weld is acceptable if : eq ysσ σ<Where: σys

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 σeq

.

β

= 0.7

for steel material with the yield strength σys

< 240 MPa

β

= 0.8

for steel material with the yield strength 240 < σys

< 280 MPa

β

= 0.85


< 340 MPa

β

= 1


< 400 MPa

Static strength assessment of fillet welds


Idealization of welds in a TIdealization of welds in a T--

butt welded joint; a) geometry and loadings, b) butt welded joint; a) geometry and loadings, b) and c) position of weld lines in the model for calculating stresand c) position of weld lines in the model for calculating stresses under ses under

axial, torsion and bending loadsaxial, torsion and bending loads

Mb

Tr

b Weldline

Weldline

b)

c)

r

2c

r

d

a)

t

d

Tr

b

Mb

h

P

;2

;2 2r

bP b

w

rT

w

M cPtd I

T r d bc orJ

σ σ

τ

⋅= =

⋅= =

r

2c2c

2c

P

b


( ) ( )22 3reqv M P T V yield IIWσ β σ σ τ τ σ= + + + ≤ −

P

M

Tr

z

x

y

a)

r

Aw

ApV

Pw

PA

σ =,

yM

w x

M cI

σ =

max

,r

rT

w CG

T rJ

τ = Vx

VQI t

τ =

τ(x,y)

x

y

CG

b)

y

z

σ(x,y)

c)

r

( )3r

ysT V P M yield AWS

στ τ τ σ σ τ= + + + < ≤ −

Combination of stress components induced by multiple loading modCombination of stress components induced by multiple loading modeses


1 2

2 cos

2 cos45

1.414

xPlt

PlhP

lhPlh

τ σ

θ

= =

=

=

=

13

31.41

.2254

1

eq

Plh

Plh

σ τ= =

≈

EXAMPLE:EXAMPLE:

Transverse fillet weld under axial loadingTransverse fillet weld under axial loading

τ1

σx

σx

P/2θ

P/2

t

h

l

P

P/2

P/2

h

AWSAWS


1

1

sin sin45sin2 2 cos45 2cos coscos2 2 cos 2

x

x

P P Plt lh lh

P P Plt lh lh

θσ σ θ

θ θτ σ θθ

= = = =

= = = =

2 22 21 13 3

2 2eqP Plh l

Ph lh

β σ τ βσ β⎛ ⎞ ⎛ ⎞= + = + =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

σ1

P/2

t

τ1

P/2

θ

IIW / ISOIIW / ISO


2) Outline of contemporary fatigue analysis 2) Outline of contemporary fatigue analysis methods methods ––

similarities, differences and similarities, differences and

limitations, limitations,

•the classical nominal stress method (S S --

NN),

• the local strain-life approach,(εε

––

NN)

• the fracture mechanics approach (da/dNda/dN

––

ΔΔKK),


•

Stress-Life Method

or the S S --

NN

approach; uses the nominal or simple engineering stress ‘ S

‘ to

quantify fatigue damage

•

Strain-Life Method

or the εε

--

NN

approach; uses the local notch tip strains εa

and stresses σa

to quantify the fatigue damage

•

Fracture Mechanics

or the da/dNda/dN

––

ΔΔKK; uses the stress intensity factor range ΔK

to quatify the fatigue crack growth rate da/dN

The Most Popular Methods for Fatigue Life The Most Popular Methods for Fatigue Life PredictionsPredictions


Information Path for Strength and Fatigue Life AnalysisInformation Path for Strength and Fatigue Life Analysis

ComponentGeometry

LoadingHistory

Stress-StrainAnalysis

Damage Analysis

Fatigue Life

MaterialProperties


Stress Parameters Used in Static Strength Stress Parameters Used in Static Strength and Fatigue Analysesand Fatigue Analyses

peakt

n

Kσσ

=

a)

σn

y

dn

0

T

ρ

σpeak

Stre

ss

S

K S a Yπ= ⋅

c)

y

a0

T

ρ

Stre

ss σπ

=Ka2

S

S

σn

y

dn

0

T

ρ

σpeak

Stre

ss

M

b)


Generation of fatigue the Generation of fatigue the SS--NN

data under data under rotational bending rotational bending

-

σn, min

+ σn,max σn,max

σn,min

timeStre

ss

(Collins, ref. 8)W


Fully reversed axial S-N curve for AISI 4130 steel. Note the break at the LCF/HCF transition and the endurance limit.

10m A maS C N N= ⋅ = ⋅

The classical fatigue The classical fatigue SS--NN

curvecurve

Number of cycles, N

Stre

ss a

mpl

itude

, Sa

(ksi

)

Low cycle fatiguePart 1

Infinite

life

Part 3

100 101 102 103 104 105 106 107 10830

40

50

60

708090

100

120

140

High cycle fatiguePart 2

Sy

Se

Su

S103


''2 22

b cf N Nf f fE

σε ε⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

Δ = +'

1

'n

E Kσσε

⎛ ⎞⎜ ⎟⎜ ⎟⎝ ⎠

= +

The StrainThe Strain--life and the Cyclic Stresslife and the Cyclic Stress--Strain Curve Obtained from Strain Curve Obtained from Smooth Cylindrical Specimens Tested Under Strain Control Smooth Cylindrical Specimens Tested Under Strain Control

((UniUni--axial Stress State)axial Stress State)

Strain -

Life Curve

εεε

log

(Δε/

2)

log (2Nf

)

σf

/E

εf

0 2Ne

σe

/E

c

b

0

Stress -Strain Curve

ε

σ

E


Determination of the Complete Fatigue Crack Growth Curve Determination of the Complete Fatigue Crack Growth Curve ((da/dNda/dN--∆∆K and K and ∆∆KKthth

))

a & b) specimens used to generate fatigue crack growth curves, c) stress history, d) fatigue crack growth rate curve.

σn

nK a Yσ π= ×W

K P a Yπ= ×

a

P

P

2a

W

a)

b)

Time

c)

P o

r σ n

0

Log(

da/d

N)

Log(ΔK)ΔKth Kmax

= Kc

m

d)

Applic

abilit

y of P

aris’

equa

tion


LOADING

F

t

GEOMETRY, Kf

PSO

MATERIAL

No N

Δσn

σe

MATERIAL

σ

ε0

E


Damage Analysis

Fatigue Life

Information path for fatigue life estimation based on the Information path for fatigue life estimation based on the SS--NN

methodmethod


Steps in Fatigue Life Prediction Procedure Based Steps in Fatigue Life Prediction Procedure Based on the on the SS--NN

ApproachApproach

The S The S ––

N methodN method

5

e) Standard S-N curves

K0K1K2K3K4K5St

ress

am

plitu

de, Δ

σ n/2

or Δ

σ hs/2

Number of cycles, N

d) Standard welded joints


a) Structure

b) Component

VP

R

Q

H

F

Weld


( )

( )

( )

( )

( )

5

5

5

5

5

11

1 5

22

2 5

33

3 5

44

4 5

55

5 5

1

1

1

1

1

m

m

m

m

m

DN C

DN C

DN C

DN C

DN C

σ

σ

σ

σ

σ

Δ= =

Δ= =

Δ= =

Δ= =

Δ= =

1 2 3 4 5 ;D D D D D D= + + + +

Δσ 1

Δσ 2

Δσ3 Δσ4Δ

σ 5

Stre

ss

t

g)

Fatigue damage:h)

Total damage:i)

Fatigue life: N

blck

=1/Dj)

Stre

ss, σ

n

f)

t

K5

σn

Steps in Fatigue Life Prediction Procedure Based Steps in Fatigue Life Prediction Procedure Based on the on the SS--NN

Approach Approach (continued)(continued)


a) Structure

Q

H

F

K 5

σn

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.

K0K1K2K3K4K5St

ress

am

plitu

de, Δ

σ n/2

or Δ

σ hs/2

Number of cycles, N0

The Similitude Concept in the The Similitude Concept in the SS--NN

MethodMethod


LOADING

F

t

GEOMETRY, Kf

PSO

MATERIAL

σ

ε0

E


Damage Analysis

Fatigue Life

Information path for fatigue life estimation based on the Information path for fatigue life estimation based on the εε--NN

methodmethod

MATERIAL½

Δε

2Nf


Steps in fatigue life prediction procedure based on theSteps in fatigue life prediction procedure based on the

εε

--

NN

approachapproach

a) Structure

b) Component


d)

σpeak

σn

σpeak

σhs

σnσhs


Steps in fatigue life prediction procedure based on theSteps in fatigue life prediction procedure based on the

εε--NN

approach approach

ε

σ

3

2,2'

4

5,5'7,7'

6

8

1,1'

Fatigue damage:

1 2 3 41 2 3 4

1 1 1 1; ; ; ;D D D DN N N N

= = = =

Total damage:

1 2 3 4 ;D D D D D= + + +

Fatigue life: Fatigue life: NN blckblck

=1/D=1/D

0

ε

σ

'1

'

n

E Kσ σε ⎛ ⎞= + ⎜ ⎟

⎝ ⎠

e)

( ) ( )'

'2 22

Δ= +

b cff f fN N

Eσε ε

σe

/E

log

(Δε/

2)

log (2Nf

)

σf

/E

εf

02Ne

2N

( )2

: = ×peakNeuberE

σσ ε

Δσpe

ak

t

1

2

3

4

5

6

7

8

1'

f)

( )

( )( )( )

' '

' '

'

'

,

,

,

?,?

Δ ΔΔ

f f

f f

f

f

fN

ε ε

σ σ

ε ε

ε μ σ

σ μ σ

ε μ σ

Δσ

σ

ε

Δε p Δε e= Δσ/Ε

Δε

0


a) Specimen

b) Notched component

εpeakx

y

z

εpeak

εpeak

( ) ( )'

'2 22

bff f fN N c

Eσε εΔ

= +

log

(Δε/

2)

log (2Nf

)

εf

0

l)

j)


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.

The Similitude Concept in the The Similitude Concept in the εε

––

NN

MethodMethod


Information path for fatigue life estimation Information path for fatigue life estimation based on the based on the da/dNda/dN--ΔΔKK

methodmethod

LOADING

F

t

GEOMETRY, Kf

PSO

MATERIAL

σ

ε0

E


Damage Analysis

Fatigue Life

MATERIAL

n

ΔKΔKth

dadN


VP

Q

R

H

F

Weld

a) Structure

b) Component


ΔS1

ΔS2

ΔS3 ΔS4

ΔS5

Stre

ss, S

t

e)

d)

Steps in the Fatigue Life Prediction Procedure Based Steps in the Fatigue Life Prediction Procedure Based on the on the da/dda/dNN--ΔΔKK

ApproachApproach


Stress intensity factor, K(indirect method)

Weight function, m(x,y)

( ) ( ), ,A

n

K x y m x y dxdy

KYa

σ

σ π

=

=

∫∫

Stress intensity factor, K(direct method)

2I yFE FE

n

K x

or

dUK E EGda

KYa

σ π

σ π

=

= =

=

σ(x, y)

f)

a

g)

( )

01

mi i i

N

f ii

i

a C K N

a a a

N N=

Δ = Δ Δ

= + Δ

= Δ

∑

∑

Integration of Paris’

equationh)

af

ai

Number of cycles , N

Cra

ck d

epth

, a

Fatigue Life

i)

Steps in Fatigue Life Prediction Procedure Based on the Steps in Fatigue Life Prediction Procedure Based on the da/dnda/dn--ΔΔKK

Approach Approach (cont(cont’’d)d)



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.

a) Structure

Q

H

F

ab) Weld detail

c) Specimena

P

P

ΔK 10-12

10-11

10-10

10-9

10-8

10-7

10-6

1 10 100

Cra

ck G

row

th R

ate,

m/c

ycle

mMPa,KΔ

The Similitude Concept in the The Similitude Concept in the da/dNda/dN

––

ΔΔKK

MethodMethod


What stress parameter is needed for the Fracture What stress parameter is needed for the Fracture Mechanics based (Mechanics based (da/dNda/dN--ΔΔKK) fatigue analysis?) fatigue analysis?

x

a0

T

ρ

Stre

ss σ

(x) ( )

2x

xKσπ

=

S

SThe 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

σ(x)

near crack tip stress or

displacement data obtained from FE analysis of a cracked body (difficult)

-

from the weight function by using the FE stress analysis data of un-cracked body (versatile and suitable for FCG analysis)


3) Load/stress histories3) 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.


Loads and stresses in a structureLoads and stresses in a structure

Load F

VP

Q

R

HF

Weld

;

;,

, ?

?i

i

ak

n

p

i

e i F F

F F

f f

g gσ

σ ⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞ ⎛ ⎞⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

= =

= =

F

t

Fi

Fi+1Fi-1

0

σn σpeak


( )

3

3

32;

; ; ;4 2 64

b b

b

eakn pM c M

SI d

W d dM L l c I

ππ

σ σ= = = = ±

= + = =Note!

In the case of smooth components,

such as the railway axle,

the nominal stress and the local peak stress are the same!

b)

d N.A.

,min 3

32n

bMd

σπ

= +

,min 3

32n

bMd

σπ

= −

1

2

3

σn,max

σn,min

timeStre

ss σn,a

1 cyclec)

1

2

3

AB

y

x

Ll

Moment Mb

a)

RARB

W/2W/2 W/2

W

The load W and the nominal stress The load W and the nominal stress σσnn

in an railway axlein an railway axle


Loads and StressesLoads and StressesThe load, the nominal stress, the local peak stress and the streThe load, the nominal stress, the local peak stress and the stress concentration factorss concentration factor

, ,

;

;

; ;

;

;

;

1

;

peaktF F

Fpeak i i

F

Fn i i

nnet

n F

Fn

Fpe

et

ak

h h KF

hk F F

k

FA

k F

k A

h F

σ

σ σ

σ σ

σ

= =

==

=

=

=

=

Axial load –

linear elastic analysis

σn

y

dn

0

T

ρ

σpeak

Stre

ss

F

F

Analytical, FEMAnalytical, FEM HndbkHndbk



, ,

,

;

;

; ;

btMpeak i pe Mi ak i i

netn

net

n M

netM Mn i i

net

M or kh M K

M cI

k M

ck k MI

σ σ

σ

σ

σ

= = ⋅

⋅=

=

= ⇒ =

Bending load –

linear elastic analysis

σn

y

dn

0

T

ρ

σpeak

Stre

ssM

b) M


peak eakt t

n

poK rS

Kσ

σσ

= =

Kt

–

stress concentration factor (net or gross, net Kt ≠

gross Kt

!!

)

σpeak

–

stress at the notch tip

σn

-

net nominal stress

S -

gross nominal stress

Stress Concentration Factors in Fatigue AnalysisStress Concentration Factors in Fatigue AnalysisThe nominal stress and the stress concentration factor in simpleThe nominal stress and the stress concentration factor in simple

load/geometry configurationsload/geometry configurations

nnet gross

P Por SA A

σ = =

grossnetn

net gross

M cM cor S

I Iσ

⋅⋅= =

Simple axial load

Pure bending load

σn

y

dn

0

T

r

σpeak

Stre

ss

S

S

TensionTension

σn

y

dn

0

T

r

σpeak

Stre

ss

MS

SM

BendingBending

net Ktgross Kt


Stress concentration factors for notched machine componentsStress concentration factors for notched machine components

(B.J. Hamrock et. al., ref.(26)

d b

h

1.0

1.4

1.8

1.2

1.6

2.0

2.2

2.4

2.6

2.8

3.0St

ress

con

cent

ratio

n fa

ctor

K

t=σ p

eak/σ

n

Diameter-to-width ratio d/b

0 0.1 0.2 0.3 0.4 0.5 0.6

σpeak

P P

σn

=P/[(b-d)h]


Stress concentration factors for notched machine componentsStress concentration factors for notched machine components

(B.J. Hamrock et. al., ref.(26)

1.0

1.4

1.8

1.2

1.6

2.0

2.2

2.4

2.6

2.8

3.0

0 0.05 0.10 0.15 0.20 0.25 0.30

H/h=6H/h=2H/h=1.2H/h=1.05H/h=1.01

Stre

ss c

once

ntra

tion

fact

or

Kt=σ p

eak/σ

n

Radius-to-height ratio r/h

peakσ

r

Mh

b

MH

2

6n

Mc MI bh

σ = =



,

,

, ;

;

;i

bti

t

F Mn i i

tpeak i

F Mpea

i

F M

k

i

ii

M

M

K

k F k

K

h

M

k F kor

hF

σ

σ

σ

= +

= ⋅ +

= +

Simultaneous axial and bending load

σn

y

dn

0

T

ρ

σpeak

Stre

ss

M

b) M

F

F

;

,

,

,

;

;

bt

tt

F M

F M

K

h

k

K

h

k From simple analytical stress analysis

From stress concentration handbooks

From detail FEM analysis


σ22

A B C

t

t

σ23

0

0

A

B

C

σ23

σ22

0

A B C

t

t

σ23

0

0

σ22

A

B

C

σ23

σ22

0

Non-proportional loading path Proportional loading path

Fσ22

σ33

ρ

2R

t

x2

x3

F

T

Tσ23 σ22

σ33

σ22

σ33

x2

x3

σ23

Fluctuations and complexity of the stress state at the Fluctuations and complexity of the stress state at the notch tipnotch tip


How to establish the nominal stress history?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, Mb

=1, T=1 in order to establish the proportionality factors, kP

, kM

, and kT

such that:

;;= ⋅ = ⋅ = ⋅P M Tn n nP M Tbk P k M k Tσ σ τ

b) The peak and valleys of the nominal stress history σn,,i

are determined by scaling the peak and valleys load history Pi

, Mb,I

and Ti

by appropriate proportionality factors kP

, kM

, and kT

such that:

, , ,;,= ⋅ = ⋅ = ⋅P M Tn i n i n i iiP M Tb ik P k M k Tσ σ τ

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.

;,, i Mi Pn b ik P k Mσ += ⋅ ⋅

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.

( ) ( ) ( )ii in P M bt tt k P k Mσ += ⋅ ⋅


Two proportional modes of loading

0

Mode a

Stre

ss σ

n,b

0

Stre

ss σ

n,a

0

Mode b

Stre

ss σ

n

Resultant stress: σn

= σn,a

+

σn,b

0

Mode a

Stre

ss σ

n,b

0

Stre

ss σ

n,a

0

Mode b

Stre

ss σ

n


(ti

)= σn,a

(ti

)+

σn,b

(ti

)

time

time

Two non-proportional modes of loading

Superposition of nominal stress histories induced by two Superposition of nominal stress histories induced by two independent loading modesindependent loading modes


How to establish the linear elastic peak stress, How to establish the linear elastic peak stress, σσpeakpeak

,,

history?history?

a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, Mb

=1, T=1 in order to establish the proportionality factors, kP

, kM

, and kT

such that:

;;peak peak peaP M Tkbk P k M k Tσ σ τ= ⋅ = ⋅ = ⋅

b) The peak and valleys of the notch tip peak stress history σpeak,,i

are determined by scaling the peak and valleys load history Pi

, Mb,I

and Ti

by appropriate proportionality factors kP

, kM

, and kT

such that:;, , ,, iipeak i peak i peak iP M Tb ik P k M k Tσ σ τ= ⋅ = ⋅ = ⋅

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.

;, ,iP Mp i bk iea k P k Mσ += ⋅ ⋅

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.

( ) ( ) ( )i ii P M bpeak t t tk P k Mσ += ⋅ ⋅


Two proportional modes of loading

0

Mode a

Stre

ssσ p

eak,

b

0

Stre

ssσ p

eak,

a

0

Mode b

Stre

ssσ p

eak


= σn,a

+

σn,b

0

Mode a

0

0

Mode b

Resultant stress: σpeak

(ti

)

= σn,a

(ti

)+

σn,b

(ti

)

time

time

Two non-proportional modes of loading

Superposition of linear elastic notch tip stress histories inducSuperposition of linear elastic notch tip stress histories induced ed by two independent loading modesby two independent loading modes

Stre

ssσ p

eak,

bSt

ress

σ pea

k,a

Stre

ssσ p

eak


Loading and stress historiesLoading and stress histories

Time

0

Stre

ss σ

peak

Time0

Load

F

Time

Fi-1

Fi

Fi+1

Fmax

0

Stre

ss σ

n

σn,i-1

σn,i

σn,i+1

σn,max

Load history, F Nominal stress history, σn Notch tip stress history, σpeak

0i-1

i

i+1

+1

-1

,n iσ

max

,

,

,

,

,ma

,

,max,

,

x

1 1

F i

t t F i

i

peak i

peak i

p

n i

n i

n in

eakni

n i

k F

K K k F

FF

σ

σ

σ σ

σ

σ

σ

σσ

=

=

= =

=

− ≤

=

≤ +

Characteristic non-dimensional load/stress history

σpeak,i

σpeak,i-1

σpeak,i+1

σpeak,max


Wind load and stress fluctuations in a wind Wind load and stress fluctuations in a wind turbine bladeturbine blade

Note! One reversal of the wind speed results in several stress reversals

Wind speed fluctuationsWind speed fluctuations

+ + Blade vibrationsBlade vibrations

→→ Stress fluctuationsStress fluctuations

Source [43]

In-plane bending Out of plane bending

Time [s]600 605 610 615 620

Time [s]600 605 610 615 620

Win

d sp

eed

[m/s

]Lo

ad-la

g st

ress

[M

Pa]

- 5

5

0

8

10

12

14

- 400

10

12

14

- 50

- 30

- 28

Win

d sp

eed

[m/s

]Lo

ad-la

g st

ress

[M

Pa]


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

Characteristic load/stress history in the aircraft wing skinCharacteristic load/stress history in the aircraft wing skin

time

Stre

ss

0

Source [9]

σσ

0

a)

LandingTaxiing

Flying

b)

c)


maxpeak

σσ

maxpeak

FF

maxpeak

σσ

maxpeak

MM

maxpeak

MM

maxpeak

GG

Stress at rear motor-car axle

Pressure in a reactor

Stress at motor-car wheel

Torsion moment at steel-mil shaft

Bending moment at stub axle of motor-car

Vertical acceleration of the Gravity Center of fighter airplane

Pressure in an oil-pipeline

Vertical acceleration of the Gravity Center of transport airplane

maxpeak

pp

maxV

V peak

GG

max1 1peak

σσ

− ≤ ≤ +

max1 1peak

FF

− ≤ ≤ +maxpeakσ +1

-1

max1 1peak

σσ

− ≤ ≤ +

max1 1peak

MM

− ≤ ≤ +

Typical load fluctuations in various engineering objectsTypical load fluctuations in various engineering objects


How to get the nominal stress How to get the nominal stress σσnn

from the from the Finite Element method stress data?Finite Element method stress data?

Notched shaft under axial, bending and torsion loadNotched shaft under axial, bending and torsion load

a)

Run each load case separately for an unit load

b) Linearize

the FE stress field for each load case

x3

Fσ22

σ33

r

D

t

x2σ22F

T

Tσ23

MMd

Discrete cross section stress distribution obtained from the FE analysis

d

σ22

0

σnx3 σpeak


How to establish the link between the fluctuating load history How to establish the link between the fluctuating load history and the stress distribution, and the stress distribution, σσ((x,yx,y)),,

in the potential crack plane?in the potential crack plane?a) The analytical or FE analysis should be carried out for one characteristic load magnitude, i.e. P=1, Mb

=1, T=1 in order to establish the link between the load and the

stress distribution, kP

, kM

, and kT

such that:

;( , ) ( , ) ( ,( , ) ; ( , ) () , )P M Tbk x yx y x y x yP k x y M k x y Tσ σ τ= ⋅ = ⋅ = ⋅

b) The fluctuating stress distributions corresponding to instantaneous peaks and valleys of the load history are determined by scaling the reference stress distributions kP

(x,y), kM

(x,y), kT

(x,y) by appropriate magnitudes of the load history Pi

, Mb,I

and Ti

such that:

Where: σ(x,y) –

stress distribution in the x-y

plane (crack plane)

kP

(x,y), kM

(x,y), kT

(x,y)

–

reference stress distributions induced by unit loads P=1, Mb

=1, T=1

;,( , ) ; (( , ) ( , ) ( ,) )), ( ,i i ii iP M Tb ik x y P k x y M kx y x y xx y y Tσ σ τ= ⋅ = ⋅ = ⋅

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.

;,( , ) ( , )( , ) ii P M b ik x y P kx y x y Mσ += ⋅ ⋅

, ,, , ;( , ) ( , )( , )n P n Mi ii n P n Mx yy k x y k xσ σσ σσ += ⋅ ⋅


How to get the resultant stress distributionHow to get the resultant stress distribution

from the from the Finite Element stress data? Finite Element stress data? (Notched shaft under axial, bending load)(Notched shaft under axial, bending load)

x3

Pσ22

σ33

r

D

t

x2Pσ23

MM

dd

σ22

0

σnbx3

Bending

d

x3

σ22

0

σnm

Axiald

σ220

σpeak

Resultant

x3

σ(x3

)


Determination of approximate stress distributions in notched Determination of approximate stress distributions in notched bodies bodies (for simultaneous axial and bending load)(for simultaneous axial and bending load)

1 3 2 42 2

1 3 2 42 2

1 1 1 1 1 1 11 1 12 3 2 2 6 22 2 4 2

1 1 1 1 1 1 11 12 3 2 2 6 22 2 4 2t

peax

xxn

kx

x x x x xr r r r

or

K x x x xr r r r

σ

σ σ

σκ

⎡ ⎤⎢ ⎥⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎢ ⎥⎣ ⎦

⎡⎢ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝ ⎠ ⎝ ⎠⎝ ⎠ ⎝ ⎠⎣

− − − −

− − − −

= + + + + + + + + −

= + + + + + + + + 1 xκ

⎤⎥ ⎛ ⎞⎢ ⎥ ⎜ ⎟⎜ ⎟⎢ ⎥ ⎝ ⎠⎢ ⎥⎦

−

κ → ∞ (for pure axial load) !

κ

= distance to the neutral axis (for pure bending load) !

pe

n

kt

aKσσ

=

x

b)

σn

dn

0

T

r

σpeak

Stre

ss

M

κ

σyy

σxx

2 4

2 4

1 1 12

1 1 12

peakyy

yynt

x x xr r

orK x x x

r r

σκ

σσ κ

σ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥

+⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥

⎣ ⎦

⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞⎢ ⎥

+⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎢ ⎥

⎣ ⎦

− −

− −

= + + −

= + + −

Note! Good accuracy for x < 3.5ρ

σnx

dn

0

T

r

σpeak

Stre

ssa) S

σyy

σxx


Cyclic nominal stress and corresponding fluctuating stress distrCyclic nominal stress and corresponding fluctuating stress distributionibution

Stre

ss σ

n

time

σn, max

σn, 0

σn, min

x3

d

σ220

Resultant

σ22

(x3

,σn,max

)σ22

(x3

,σn,0

)σ22

(x3

,σn,min

)


Loads and stresses in a welded structureLoads and stresses in a welded structure

σn

Load F

σpeak


Removing material from a clay mine in TennesseeRemoving material from a clay mine in Tennessee


Stress history a3799_01 (Right Hand Axle Torque)

–

as recorded

Stress history a3799_01 –

with removed ranges less than 5% of the largest one in the history

The complete Right Hand Axle Torque stress/load historyThe complete Right Hand Axle Torque stress/load history


Details of the stress history a3799_c01; visible repeatable workDetails of the stress history a3799_c01; visible repeatable working cycles; ing cycles; ((Right Hand Axle Torque history Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the histowith removed ranges less than 5% of the largest one in the history)ry)

Smax,2

Smax,3

Smax,4Smax,i

Smax,1 Smax,1 =


ReliaSoft Weibull++ 7 - www.ReliaSoft.com

Probability - Lognormal

m=4.1795, s=0.1095, r=0.9831 (a3799_01)

Stress, (MPa)

Unr

elia

bilit

y, F

(t)

10.000 100.0000.100

0.500

1.000

5.000

10.000

50.000

99.900

0.100

Probability-Lognormal

Max_01Lognormal-2PRRX SRM MED FMF=87/S=0

Data PointsProbability Line

Grzegorz GlinkaUniversity of Waterloo22/05/20077:24:10 AM

mσ

= 65.72 MPa

μσ

= 7.21 MPa

COV=0.1098

Distribution of peak stresses Distribution of peak stresses SSmaxmax

in the stress history a3799_01; in the stress history a3799_01; ((Right Hand Axle Torque History)Right Hand Axle Torque History)


NonNon--dimensional stress history a3799_01; dimensional stress history a3799_01;

((Right Hand Axle Torque history Right Hand Axle Torque history with removed ranges less than 5% of the largest one in the histowith removed ranges less than 5% of the largest one in the history)ry)

Statistical data for the scaling peak stress Smax

Log-Normal Probability distribution, LN (4.1975, 0.1098)

Mean Value Standard Deviation Coefficient of variation

mσ

= 65.72 MPa

μσ

= 7.21 MPa

COV=0.1098 (actual)

mσ

= 1 μσ

= 0.1098 COV=0.1098 (Non-dimensional)


The The RainflowRainflow

Cycle Counting ProcedureCycle Counting Procedure


Loads and stresses in a structureLoads and stresses in a structure

σn

Load F

σpeak


Stress/Load Analysis Stress/Load Analysis --

Cycle Counting ProcedureCycle Counting Procedureand Presentation of Resultsand 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. 141.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.

Unfortunately, the cycle definition is not simple in the case of

a variable amplitude signal. The only non-

dubious quantity, which can easily be defined, is a reversal, example of which is marked with the thick line in the Figures below.

max min

max min

max min

2

2

a

m s

stress range

stress amplitude

mean stres

σ σ σ

σ σσ

σ σσ

Δ = − −

−= −

+= −

σmax

σmin

timeStre

ssσa

1 cycle

0

σm ∆σ


Bending Moment Time SeriesBending Moment Time Series

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

20

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81

Load point No.

Ben

ding

mom

ent

[10×

kNm

]

Bending Moment measurements obtained at constant time intervals


Bending Moment History Bending Moment History --

Peaks and ValleysPeaks and Valleys

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

20

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Load point No.

Ben

ding

mom

ent v

alue

[10×

kNm

]

Bending Moment signal represented by the reversal point values


Constant and Variable Amplitude Stress Histories;Constant and Variable Amplitude Stress Histories; Definition of the Stress Cycle & Stress ReversalDefinition of the Stress Cycle & Stress Reversal

max min

max min

max min

min

max

;

2 2

;2

a

m

R

σ σσ

σ

σ σ σ

σσ

σ

σσ

+=

=

Δ = −

−Δ= =

Stre

ss

Time0

Variable amplitude stress history

One reversal

b)

0

One cycle

σmean

σmax

σmin

Stre

ss

Time0

Constant amplitude stress historya)


Stress Reversals and Stress Cycles in a Variable Stress Reversals and Stress Cycles in a Variable Amplitude Stress HistoryAmplitude Stress History

The reversalreversal

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 rainflowrainflow

cycle counting method has been accepted world-wide as the most appropriate for extracting stress/load cycles for fatigue analyses. The rainflowrainflow

cycle is defined as a stress excursion, which when applied to a deformable material, will generate a closed stressstress--strain hysteresis loopstrain 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 rainflowrainflow

counted cyclescounted cycles

content is shown in the following Figure.


A rainflow

counted cycle

is identified when any two adjacent reversals in the stress history satisfy the following relation:

1 1i i i iABS ABSσ σ σ σ− +− ≤ −

Stre

ss

Time

Stress history Rainflow

counted cycles

σi-1

σi-2

σi+1

σi

σi+20

Stress History and the Stress History and the ““RainflowRainflow””

Counted CyclesCounted Cycles


A rainflow

counted cycle

is identified when any two adjacent reversals in thee stress history satisfy the following relation:

1 1i i i iABS ABSσ σ σ σ− +− < −

The stress amplitude of such a cycle is:

1

2i i

a

ABS σ σσ − −

=

The stress range of such a cycle is:

1i iABSσ σ σ−Δ = −

The mean stress of such a cycle is:

1

2i i

mσ σσ − +

=

The Mathematics of the Cycle The Mathematics of the Cycle RainflowRainflow

Counting Counting Method for Fatigue Analysis of Stress/Load HistoriesMethod for Fatigue Analysis of Stress/Load Histories


The ASTM The ASTM rrainflowainflow

cycle counting procedurecycle counting procedure

--

exampleexample

Determine stress ranges, ΔSi

, and corresponding mean stresses, Smi

for the stress history given below. Use the ASTM ‘rainflow’

counting procedure.

Si

= 0, 4, 1, 3, 2, 6, -2, 5, 1, 4, 2, 3, -3, 1, -2

(units: MPa·102)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

1

2

3

4

5

6

0

-1

-2

-3

Stre

ss S

i(M

Pa·1

02)

Reversing point number, i


Reversal point No.

RainflowRainflow

cycle counting cycle counting ––

The The Direct Direct MMethod:ethod:1. Find the absolute maximum revesal point; 1. Find the absolute maximum revesal point;

2. Add the absolute maximum at the end of the history, 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); 3. Start counting form the reversal no. 3 (always);

Stress History

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Stre

ss (M

Pa)x

102

Absolute maximum

-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point


-3-2-10

1

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point

4 33 4 3 4 ,3 4

3 23 2 1; 2.5;2 2mσ σσ σ σ σ− −

+ +Δ = − = − = = = =

1 21 2 1 2 ,1 2

4 14 1 3; 2.5;2 2mσ σσ σ σ σ− −

+ +Δ = − = − = = = =


10 1110 11 10 11 ,10 11

2 32 3 1; 2.5;2 2mσ σσ σ σ σ− −

+ +Δ = − = − = = = =

8 98 9 8 9 ,8 9

1 41 4 3; 2.5;2 2mσ σσ σ σ σ− −

+ +Δ = − = − = = = =

-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point

-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point


-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point

7676 7 6 ,6 7

2 52 5 7; 1.5;2 2mσ σσ σ σ σ− −

+ − +Δ = − = − − = = = =

-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point

13 1413 14 13 14 ,13 14

1 21 ( 2) 3; 0.5;2 2mσ σσ σ σ σ− −

+ −Δ = − = − − = = = = −


-3-2-101

23

456

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Stre

ss (M

Pa)x

102

Starting point

5 1255 12 12 ,5 12

6 36 ( 3) 9; 1.5;2 2mσ σσ σ σ σ− −

+ −Δ = − = − − = = = =

Cycles counted Cycles counted ––

Direct methodDirect method1. ∆σ3-

4 =1; σm,3-

4 = 2.5; 2. ∆σ1-

2 =3; σm,1-

2 = 2.5; 3. ∆σ10-

11

=1; σm,10-

11

= 2.5; 4. ∆σ8-

9 =3; σm,8-

9 = 2.5; 5. ∆σ6-

7 =7; σm,6-

7 = 1.5; 6. ∆σ13-

14 =3; σm,13-

14

=-0.5; 7. ∆σ5-

12 =9; σm,5-

12 = 1.5;


Extracted rainflow cycles, Extracted rainflow cycles, ΔσΔσ--

ΔσΔσmm


EExtracted rainflow cyclesxtracted rainflow cycles

––

the the ΔσΔσ--

ΔσΔσmm

matrixmatrix

Total number of cycles, N=854

Δσ/σm -32 -22 -13 -3.2 6.44 16.1 25.7 35.3 45 54 64.1 73.7 83.3 92.9 103 112 122 131 141 151 Δσ298.8 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1283.9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0268.9 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1254 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1239 0 0 0 0 0 0 0 1 2 2 0 0 0 0 0 0 0 0 0 0 5

224.1 0 0 0 0 0 0 0 0 2 2 1 0 0 0 0 0 0 0 0 0 5209.2 0 0 0 0 0 0 0 3 4 5 2 0 0 0 0 0 0 0 0 0 14194.2 0 0 0 0 0 1 0 1 7 2 0 0 0 0 0 0 0 0 0 0 11179.3 0 0 0 0 0 0 1 0 4 4 0 0 0 0 0 0 0 0 0 0 9164.3 0 0 0 0 1 0 0 0 3 1 0 0 0 0 0 0 0 0 0 0 5149.4 0 0 0 0 0 0 1 0 0 0 0 2 1 0 0 0 0 0 0 0 4134.5 0 0 0 0 0 0 0 0 0 0 0 4 1 1 0 0 0 0 0 0 6119.5 0 1 1 0 0 0 0 0 0 0 3 1 5 1 2 0 0 0 0 0 14104.6 0 0 1 2 1 0 0 0 0 2 4 3 7 3 2 1 2 1 0 0 2989.64 0 1 2 3 7 2 0 0 0 1 2 8 10 7 5 6 2 1 0 0 5774.7 1 1 3 4 3 5 0 1 2 2 10 18 23 20 17 11 4 1 0 0 126

59.76 2 1 5 7 4 1 4 5 1 2 11 20 34 31 31 28 9 7 1 1 20544.82 1 6 9 7 9 7 10 3 3 8 15 37 49 64 62 41 16 11 2 1 36129.88 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 014.94 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

854

Mean stress, σm

Stre

ss ra

nge,

Δσ


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 nj

/NT

vs. Δσj

/Δσmax

co-ordinates. The experimental probability density, f(xj

), for stress range, Δσj

, is determined as:

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:

( ) jj

T

nf x

N=

( )1

i ji

ji T

nF x x

N

=

=

< = ∑


1 298.8 12 283.86 03 268.92 14 253.98 15 239.04 56 224.1 57 209.16 148 194.22 119 179.28 910 164.34 511 149.4 412 134.46 613 119.52 1414 104.58 2915 89.64 5716 74.7 12617 59.76 20518 44.82 361


b) the stress range b) the stress range frequency distribution frequency distribution diagramdiagram

Number cycles N

Stre

ss ra

nge

Δσ j

j=7j=6

nj=6

-

number of cycles Δσj=6in the class j=6

Total number of cycles in the entire history,

NT

Δσmax

Rel

ativ

e st

ress

rang

e Δ

σ j/ Δ

σ max

Relative number of cycles N/NT

0

j=7j=6j=4321

nj=6

/NT

Δσmax

/Δσmax1

Δσj=4

/Δσmax

0.5 1.0

0.5

0

j=4321

a) The stress range a) The stress range exceedanceexceedance

diagram diagram (stress spectrum)(stress spectrum)


a) Probability Diagram Obtained from the Experimental a) Probability Diagram Obtained from the Experimental Cumulative Spectrum; b) Cumulative Probability Cumulative Spectrum; b) Cumulative Probability F(xF(x<<xxjj

))

Relative stress range, xj

= Δσj /Δσmax

Prob

abili

ty f(

x j)=

n j/N

T

0j=7 j=6 j=4

0.3

0.5 1.0

0.1

0.2

j=5 j=3 j=2 j=1

a)

0j=7 j=6 j=4

0.5 1.0

0.25

j=5 j=3 j=2 j=1

0.50

0.75

1.00

nn

nn

nn

NT

76

54

32

++

++

+

nn

nn

nNT

76

54

3+

++

+

nn

nn

NT

76

54

++

+


= Δσj /Δσmax

Cum

ulat

ive

prob

., F(

x<x j

)=Σn

j/NT b)

Continuous probability distribution function: p(x), p(∆σ/ ∆σmax

), p(Sa

/Sa,max

)…


a) Probability a) Probability DensityDensity

Diagram obtained from the Diagram obtained from the ExperExper. . Cumulative Spectrum; b) Cumulative Probability Cumulative Spectrum; b) Cumulative Probability F(xF(x<<xxjj

))

Continuous probability density distribution: p(x), p(∆σ/ ∆σmax

), p(Sa

/Sa,max

)…

( ) ( )

( )( )

<

= =Δ

< + Δ =

Δ =

Δj j T

j j j

j j j T

jj j

f x n Np ;

P x x x x

p x x

x

n

x

N

x

;

0j=7 j=6 j=4

0.5 1.0

0.25

j=5 j=3 j=2 j=1

0.50

0.75

1.00

nn

nn

nn

NT

76

54

32

++

++

+

nn

nn

nNT

76

54

3+

++

+

nn

nn

NT

76

54

++

+


= Δσj /Δσmax

Cum

ulat

ive

prob

., P(

x<x j

)=Σn

j/NT ( ) ( )

1 1

< = Δ =∑ ∑j j

j j j j TP x x p x x n N


= Δσj /Δσmax

Prob

abili

ty d

ensi

ty, p

(xj)

0j=7 j=6 j=4

0.3

0.5 1.0

0.1

0.2

j=5 j=3 j=2 j=1

Δx6 Δx5

p(x 7

)=f(x

7)/∆x

7=n 7/N

T/∆

x 7


The use of the probability density distributionThe use of the probability density distribution

Given are:

The maximum stress amplitude, Sa,max

and the total number of cycles in the spectrum NT

and the probability density function p(Sa

/Sa,max

) = p(x);

The task

of the analyst is to determine the number of applications (cycles) nj

of

stress amplitudes within

the interval (Sa,j

< Sa

< Sa,j+1

);

Find

how many cycles nj with amplitudes Sa,j

< Sa

< Sa,j+1

are to be found out in the total population

of NT

load cycles to be applied.

xj

=Sa,j

/Sa,max

and

xj+1

=Sa,j+1

/Sa,max.

1+

= ⋅ ∫j

j

x

j Tx

n N p(x)dx

Number of cycles in the class j:

12

+ += a,j a,ja,j

S SS

Average stress in the class j:


Various Various LoadLoadinging

SpectraSpectraa) constant amplitude, b) mixed spectrum, c) Gaussian normal spectrum, d-

e) mixed spectra

Relative number of cycles, Ni

/NTotal

Rel

ativ

e st

ress

am

plitu

de, σ

a,i/

σ max

a

σ(t)

σ(t)

d

f

σ(t)

t

t

t10-6 10-5 10-4 10-3 10-2 10-1 10

0.2

0.2

0.2

0.2

1.0

0

a

bc

d

e

f


Stre

ss M

PaS

tress

am

p. M

Pa

Stress signal

–

Time Domain RepresentationTime Domain Representation

Stress signal

–

Frequency Domain RepresentationFrequency Domain Representation

Sequence of peaks and valleys

Amplitudes associated with certain frequencies

Courtesy of D. Socie, Univ. of Illinois

Ways of representing the stress signalWays of representing the stress signal


A Frequency Domain approach to fatigue A Frequency Domain approach to fatigue analysis of random cyclic stress processes 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.


Load time Load time historyhistory

Structural Structural modelmodel

General fatigue analysis proceduresGeneral fatigue analysis procedures

Stress time Stress time historyhistory

Stress CyclesStress CyclesMiner Miner

fatigue fatigue damagedamage

FATIGUE FATIGUE LIFELIFE

Acceleration Acceleration PSDPSD

Transfer Transfer functionfunction

( TIME DOMAIN )( TIME DOMAIN )

( FREQUENCY DOMAIN )( FREQUENCY DOMAIN )

Stress Stress PSDPSD


RANDOM PROCESS ANALYSIS

p(∆S)

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

Courtesy of A. Halfpenny, nCode Int.


The technique was pioneered by electronics engineers in the 1940s. 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..

0 5

5

5

Extract time signal

0 5

1

Increasing filter frequency

frequency

Low pass filter over a range of frequencies

0 5

5

52

Square filtered time signal to remove negatives

0 55

5

Calculate mean value (i.e. mean square

amplitude)

0 5

20

frequency

PSD = change of mean2

vs. frequency

0 5

1

frequency

Plot cumulative mean2

vs. filter frequency

WhatWhat’’s a PSD?s a PSD?


Descriptors of random processesDescriptors of random processes

t

X(t)

0

T

Root Mean Square (RMS)Root Mean Square (RMS)

[ ]

( )

2

2

0

1 T

RMS

X t

x p(x)dx

dtT

+∞

−∞

⎡ ⎤⎣ ⎦

= ⋅

≅

∫

∫

2

1

TN

ii

T

xRMS

N==∑

or

0

t

Xi

NT

The RMS is equal to the standard deviation σx

when the mean value is zero, i.e. when μx

=0!


Descriptors of random processes Descriptors of random processes ( continued)( continued)

The meaning and magnitude of the RMS parameterThe meaning and magnitude of the RMS parameter

Δx

xmax

X(t)

xmax

X(t)

0

0

xmax

X(t)

0

t

t

t

RMSμx

xmin

xmax

xmin

xmin

RMS=0.707xmax

μx

=0.637xmax

Δx=|xmax

-xmin

|=2xmax

= 2xa

Crest factor =xmax

/RMS=1.414

RMS=xmax

μx

=xmax

Δx=|xmax

-xmin

|=2xmax

= 2xa

Crest factor =xmax

/RMS=1

RMS=0.577xmax

μx

=0.5xmax

Δx=|xmax

-xmin

|=2xmax

= 2xa

Crest factor =xmax

/RMS=1.733


frequency Hzfrequency Hzfrequency Hzfrequency Hz

Various time histories and their Various time histories and their PSDsPSDs



The NatThe Natuure of Frequency Analysis re of Frequency Analysis –– the light spectrum analogythe light spectrum analogy


•

Fourier Transform

is the mathematical equivalent


What is What is thethe

Power Spectrum Density (PSD) ? Power Spectrum Density (PSD) ?



TheThe

Fourier TransformFourier TransformAfter replacing the ‘n/T’

term by frequency ‘f’

the Fourier transform equations can be written as:

( ) ( )+∞

− ⋅ ⋅ ⋅

−∞

= ∫ 22 i f tX f X t e dtπ ( ) ( )+∞

− ⋅ ⋅ ⋅= ∫ 2

0

2 i f tX t X f e dfπ

Fourier Transform

from time to frequency domain:

Inverse Fourier Transform

from frequency to time domain:

( ) ( )⎛ ⎞

− ⋅ ⋅⎜ ⎟⎝ ⎠= ⋅∑

22 T

ni kN

n kkT

TX f X t eN

π

( ) ( )⎛ ⎞

− ⋅ ⋅⎜ ⎟⎝ ⎠= ⋅∑

21 T

ni kN

k kn

X t X t eT

π

ContinuousContinuous

process

Discrete processDiscrete process

Inverse Fourier TransformInverse Fourier Transform

Fourier TransformFourier Transform

Time Time domaindomain

Frequency Frequency domaindomain


The Number of CyclesThe Number of Cycles

t

Stre

ss

0

1 sec

P

0+

Upward zero crossing, 0+

Peak P

( )( )

EIF

E P

+

= =0 3

5

SO Rice, 1954

E[0+]

Expected number of upward zero crossing 0+

E[P]

Expected number of peaks

( )= ⋅ ⋅ δ∑ nnm f G f f

Number of upward zero crossing: ( ) 0

2

mE 0

m+ =

Number of peaks: ( ) 4

2

mE P

m=

The nth

moment area under PSD:

Frequency f, [Hz]

G(f)

, (St

ress

2 )/H

z

fk

Gk(f

)

δf

Irregularity factor


The Cycle Ranges The Cycle Ranges –– narrow band narrow band processprocess

JS Bendat, 1964 Courtesy of A. Halfpenny, nCode Int.

( )Δ

−ΔΔ =

2

08

04

SmSp S e

m

N(∆S)∆S

(∆S)2


The The Cycle rangesCycle ranges

––

widewide

band band processprocess

(Method of T. Dirlik, 1985)

p(S)D

=f(m0

, m1

, m2

, m4

)

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