© 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 ≤
1
© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
log NNRi
log Δσ
nEi
γFf Δσi
given stress range i
given number of cycles i
number of cycles up to collapse
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
Complementary notes
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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
Eurocode EN 1993-1-9 (Fatigue)
1 - detail category ∆σC2 - constant amplitude fatigue limit ∆σD3 - cut-off limit ∆σL
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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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© Prof. Ing. Josef Macháček, DrSc.
2C08 - 3
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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