Damage due to fatigue occurs when loading is markedly varying in time. R decreases with time S T. MSÚ F max

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5. Fatigue of steel structures

Fatigue loading, Wöhler’s approach and fracture mechanics, fatigue strength, influence of notches, damage accumulation, Eurocode approach.

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

Resistance R decreases due to:

- initiation of cracks, - cracks growth.

Fatigue limit state (in general): Fmax(T) ≤ Rmin(T) (valid for given time T) for required probabilities p

S₀

T₀

S_T R decreases with time

MSÚ F_max= R_min loading F, resistance R reliability S [ % ] time T

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Fatigue tests

(see mechanical tests of material, bachelor course)

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

design fatigue strength curve – e.g. for surviving with probability of p = 95 % (hyperbola) „time strength" (for N_i cycles) cut-off limit (permanent fatigue strength) _N

i [ N ] number of cycles up to damage

Δ

σ

Wöhler’s curve

σ

1 cycle stress range Ncycles (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 Δσ

a

N = logN =loga−mlogΔσ

log Δ

σ

log N N = 2 ×10 6 N = 5 ×10 6 N = 1 ×10 8 bilinear trilinear designation of category Δσ_C

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 elements.

• Real steel structure (e.g. various welded pieces): - time to initiation of cracks is very short,

- fatigue strength (Δ

σ

R) is given especially by time of crack propagation up to „critical 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

ϕ

_fat (given in standards).

σ

T

In fatigue design may be used:

1. Constant amplitude of stress range

Δ

σ

N

Δ

σ

and N are approximately estimated.

In Eurocodes is determined „equivalent stress range“ Δ

σ

E,2, which corresponds to fatigue damage of N = 2×106:

Δ

σ

_E,2=

λ

₁

λ

₂

λ

₃... Δ

σ

_k

product of equivalent damage factors

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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: Δ

σ

₁ Δ

σ

2 Δ

σ

3 Δ

σ

4 1 1 2 3 4 5 2 3 4 5 1 2 3 4 5 4' 3' history after filtration idea of "pagoda" (turned of 90º)

The stress ranges are arranged into several degree spectrum (for several Δ

σ

):

N Δ

σ

n₁(for stress range Δσ₁)

n₂ n₃ histogram: Δ

σ

Δσ1 Δσ₂ Δσ₃ Δσ₄ N n₁ n₂ n₃ n₄

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Determination of fatigue strength

• Influence of stress range Δ

σ

. is substantial. • The fatigue strength in compression is higher:

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

• Influence of stress concentration is essential:

• Influence of yield point f_yis 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% of6 Δσonly)

NOTCHES are concentrators of stresses → cracks,

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Solution of fatigue problems

1. 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:

γ

_Ff= 1,00 Fatigue resistance:

according to assessment method

- damage tolerant method (requires inspections, maintenance):

γ

_Mf= 1,15 - safe life method (without inspections):

γ

_Mf= 1,35

(the coefficients may be lowered for elements with lower consequences)

The design may be performed for:

• constant amplitude of nominal (equivalent) stress range Δ

σ

_E,2, • stress range spectrum.

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Design for „constant equivalent stress range amplitude"

For direct stresses:

Mf C E,2 Ff

Δ

γ

σ

γ

≤

stress range of equivalent nominal stress

(must be < 1,5 f_y, including dynamic coefficient

ϕ

_fat)

„fatigue strength" for 2.106_cycles

given by name of detail category (similarly for shear)

Detail categories

Curves in log scale: DC 36, 40, ... 140, 160 3

Δσ

a

N

=

5

Δσ

a

N

=

const. 160 160 cut-off limit

cut-off limit for

„constant amplitude"

117

2.106

5.106 108 cycles N

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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 k_s.

Design for „stress range spectrum"

For several degree spectrum (Δ

σ

_i, n_i, see e.g. for i = 4) the Palmgren-Miner linear damage accumulation hypothesis may be used:

1 ≤

∑

=

n i _Ri Ei d

N

n

D

number of cycles with amplitude

γ

_FfΔσ_i

number of cycles with the same amplitude up to collapse, determined from curve corresponding to

category of given detail

log N N_Ri

log Δ

σ

n_Ei

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Recommendations for fatigue design:

1. Selection of suitable details (to minimize notches).

2. Restriction of tension residual stresses (Ö welds of necessary size only, multilayer welds are better). 3. Correct determination of fatigue loading (Δ

σ

, N).

Fabrication:

1. Without notches (possibly grinding, TIG remelting, trimming by mechanical way - by hammering, shot peening; in progress ultrasonic +

mechanical treatment). 2. Low residual stresses (MAG, TIG welding).

Example of crane girder:

t max. 100 manual weld: KD 100 MAG, SAW: KD 112 KD 80 KD 80 older opinions,

today frequently welded ÖKD 80

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Complementary notes:

Fracture mechanics

Unlike as in Wöhler’s approach the development of given crack is investigated. 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-cycle

fatigue (crack vicinity is plastic). Linear fracture mechanics

σ r→ 0 2a b r K

π

σ

2 I max =

coefficient of stress intensity (after Irwin).

K_Imay be determined by FEM.

Solution consists of:

b) (a, f a K_I =σ π a) Stress in crack face:

b) Velocity of crack spread (Paris’ law): m d d K C N a ₌ _Δ N number of cycles C, m material constants

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For given K_I= K_IC („fracture toughness – material constant) a „critical length“ of the crack a_cr may be determined:

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

2 b) , cr (a c cr 1 ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ = f K a σ π I ( )

∫

=

cr 0 ΔK

d

a a

f

a

N

log Δσ log N approx. 10 000 cycles quasi-static fracture low-cycle fatigue multi-cycle fatigue cut-off limit

Nonlinear fracture mechanics (low-cycle fatigue)

Region of plastic deformations – use of Δε_plnecessary. Energy of deformation is determined by J integral.