3. Fatigue of steel structures. -...

© Prof. Ing. Josef Macháček, DrSc.

2C08 - 3

3. Fatigue of steel structures.• Fatigue loading, Wöhler’s approach and fracture mechanics, fatigue strength,

influence of notches, damage accumulation, Eurocode approach.

S0

T0

STR decreases with time

ULSFmax = Rmin

loading F,resistance R

reliabilityS [ % ]

Resistance R decreases due to:- initiation of cracks,- cracks growth.

Fatigue limit state (in general):

Damage due to fatigue occurs when loading is markedly varying in time.

(valid for given time T)

for required probabilities p

)m in()m ax( TT RF ≤

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low-cycle fatigue (< about 50 000 cycles, plastic behaviour)

multi-cycle fatigue (elastic behaviour)

design fatigue strength curve – e.g. for survivingwith probability of p = 95 %(hyperbola)

„time strength"(for Ni cycles)

cut-off limit(permanent

fatigue strength) Ni [ N ] number of cycles up to damage

∆σ

Wöhler’s curve

Fatigue tests (see mechanical tests of material, bachelor course)

σ

1 cycle

Δ σstress range

N cycles (time)

regimes:

pulsating tension

alternating loading

pulsating compression

+ σ

+-

-

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Wöhler’s curve in log coordinates (S-N curves, stress-number of cycles curves):

Usually expressed in the form:

i.e.m∆σ

aN = σ∆logloglog maN −=

log Δσ

log N

bilineartrilinear

designationof categoryΔσC

Fatigue is predominantly investigated experimentally. Cardinal difference is in behaviour of:

• Machined specimen (e.g. as in tensile test):- decisive is initiation of cracks (due to pores, defects): important for mechanical(machined) elements.

• Real steel structure (e.g. various welded pieces):- time to initiation of cracks is very short,- fatigue strength (ΔσR) is given especially by time of crack propagation up to „critical crack length“ (fatigue fracture).

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Determination of loading effects

Actual loading has stochastic distribution.

Dynamic effects are taken into account:- by dynamic calculations,- approximately with help of dynamic coefficient ϕfat (given in standards).

σ

T

In fatigue design the following loading simulation may be used:

1. Constant amplitude of stress range

Δσ

N

Here Δσ and N are approximately estimated.

In Eurocodes is determined „equivalent stress range“ΔσE,2, which corresponds to fatigue damage of N = 2×106 :

ΔσE,2 = λ1 λ2 λ3 ... Δσk

product of equivalent damage factors (for bridges and cranes are given in Eurocodes)

characteristic stress range

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2. Stress range spectrum

Actual stress distribution is evaluated by some of the cycle counting methods, e.g.:- reservoir method:

- rainflow method:

Δσ1

Δσ2 Δσ3 Δσ4

1 1

2

3

4

5

2

3

4

5

1

23

454'3'

history afterfiltration

idea of "pagoda"(turned by 90º)

The stress ranges are arranged into several degree spectrum (for several Δσ):

N

Δσ n1 (for stress range Δσ1)n2

n3histogram:

ΔσΔσ1

Δσ2 Δσ3 Δσ4

Nn1 n2 n3 n4

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Determination of fatigue strength• Influence of stress range Δσ is substantial.

• The fatigue strength in compression is higher:

ATTENTION: welded elements have always tensionresidual stresses in weld location !!! → always tension.

• Influence of stress concentration is essential:

• Influence of yield point fy is negligible(steel S235 and S460 have roughly the same fatigue strength).

• Influence of environment: fatigue strength is lowered by aggressive environment, corrosion, low and high temperatures.

+

-

Δσ

Δσ ( in compression may be taken 60% of Δσ only)

NOTCHES are concentrators of stresses → cracks,

they are especially at weld locations (see detail categories).

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Solution of fatigue problems1. Wöhler’s approach (for design of new structures – standards, Eurocode).

2. Fracture mechanics: Investigates development of a crack → enables to determine „residual life".

Fatigue design according to Eurocode (EN 1993-1-9)Loading: design values of stress range for: γFf = 1,00

Fatigue resistance:according to assessment method- damage tolerant method (requires inspections, maintenance): γMf = 1,15- safe life method (without inspections): γMf = 1,35

Note: the coefficients may be lowered for elements with lower consequences (for 1.00 and 1,15).

The design may be performed for:a) Constant amplitude of nominal (equivalent) stress range ΔσE,2.b) Stress range spectrum.

(fatigue)

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a) Design for „constant equivalent stress range amplitude"

For direct stresses:Mf

CE,2Ff

∆∆γσ

σγ ≤

stress range of equivalent nominal stress(must be < 1,5 fy, including dynamic coefficient ϕfat)

„fatigue strength" for 2.106 cyclesgiven by name of detail category

(similarly for shear)

Detail categories:

see fatigue curves in ”Complementary notes”,

(independent of steel grade)

Curves in log scale:DC 36, 40, ..... 140, 160

3∆σaN =

5∆σaN =

160 (= ΔσC)

cut-off limit

cut-off limit for„constant amplitude" 117(= ΔσD)

2.1065.106 108 cycles N

ΔσR

this curve is at the picture:the number corresponds toΔσR = ΔσC for 2·106 cyclesvalid for DC 160

const.

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log NNRi

log Δσ

nEi

γFf Δσi

given stress range i

given number of cycles i

number of cycles up to collapse

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Modifications of the assessment:

- compressive portion of the stress range may be reduced to 60 %,- due to size effect (usually t > 25 mm) the fatigue strength is reduced by coefficient ks.

b) Design for „stress range spectrum"For several degree spectrum (Δσi, ni, see e.g. figure in page 4 for i = 4) the Palmgren-Miner linear damage accumulation hypothesis may be used:

1≤∑=n

i Ri

Eid N

nD

number of cycleswith amplitude γFf Δσi

number of cycles with the same amplitude up to collapse, determinedfrom curve corresponding to category of given detail.

Note: calculation of NRi for given DCis presented in Complementary notes.

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Recommendations for fatigue design:1. Selection of suitable details (to minimize notches).2. Restriction of tension residual stresses ( welds of necessary size only,

multilayer welds are better).3. Correct determination of fatigue loading (∆σ, N).

Recommendation for fabrication:1. Increased quality of fabrication: e.g. execution class EXC3.2. Without notches (possibly grinding, TIG remelting, trimming by mechanical way

- by hammering, shot peening (in progress is ultrasonic + mechanical treatment).1. Low residual stresses (MAG, TIG welding).

Example of a crane girder:

t

max.100

manual weld: DC 100MAG, SAW: DC 112

DC 80

DC 80

older opinions,today frequently welded

DC 80

DC 90DC 40

r ≥ 150

r

DC from 40 to 90

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Complementary notes

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Eurocode EN 1993-1-9 (Fatigue)

1 - detail category ∆σC2 - constant amplitude fatigue limit ∆σD3 - cut-off limit ∆σL

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Note for a use of Palmgren-Minerova hypothesis:

Relations between fatigue strengths for 2·106 cycles (ΔσC, i.e. name of the DC), 5·106 cycles (ΔσD) and 108 cycles (ΔσL) according to figures at pages 8 and 12 are:

ΔσD = 0.737 ΔσCΔσL = 0.549 ΔσD

Therefore the limit number of cycles Ni corresponding to stress range γFf Δσi and given detail category (expressed by value of ΔσC) is:

36102 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

iMfFf

CNσΔγγ

σΔi

56105 ⎟⎟

⎠

⎞⎜⎜⎝

⎛⋅=

iMfFf

DNσΔγγ

σΔi

∞=iN

… for γFf γFM Δσi ≥ ΔσD

… for ΔσD > γFf γFM Δσi ≥ ΔσL

… for ΔσL > γFf γFM Δσi

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Fracture mechanics

Unlike as in Wöhler’s approach the development of given crack is investigated. This enables to determine residual life of the structure.

1. Linear fracture mechanics - investigates the crack within multi-cycle fatigue (most of the body is elastic).

1. Nonlinear fracture mechanics - investigates the crack within low-cyclefatigue (crack vicinity is plastic).

Linear fracture mechanicsσ

r → 02a

b

rKπ

σ I2max =

coefficient of the stress intensity (after Irwin). For a givenmodel of a crack may be determined numerically by FEM.

Solution consists of:

b)(a,faK πσ=Ia) Stress in crack face:

b) Velocity of crack spread (Paris’ law): mdd KCNa Δ=

N number of cycles,C, m material constants,ΔK amplitude KI : i.e. (KI ,max- KI,min)/2.

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For given KI = KIC (where KIC is „fracture toughness“ = material constant)a „critical length“ of the crack acr may be determined:

- and finally by integration of Paris’ law also the residual life(i.e. the number of cycles up to fatigue damage):

2

b),(a

ccr

cr

1⎟⎟⎠

⎞⎜⎜⎝

⎛=

fKa

σπI

( )∫=cr

0 ∆K

da

a faN

log Δσ

log Napprox. 10 000 cycles

quasi-static fracturelow-cycle fatigue

multi-cycle fatiguecut-off limit

Nonlinear fracture mechanics (for low-cycle fatigue)

For region of plastic deformations use of Δεpl is necessary.To determine energy of deformation the J integral is used.

σ

ε

εpl εel

εtot

Manson-Coffin relation:

Manson relation:

( )C, N2pl εεΔ = 2N number of half-cyclesC constant (-0,5 up to - 0,8)ε' 0,5 up to 0,7εyfy' coeff. of fatigue strength ≈ fyCb'

y N'NE/f )2()2)((pleltot εεΔεΔεΔ +=+=

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3. Fatigue of steel structures. -...

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