Structures 2

7/29/2019 Structures 2

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© 5 Prof. Ing. Josef Macháček, DrSc.

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5. Fatigue of steel structuresFatigue 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): )Tmin()Tmax( RF ≤ (valid for given time T)

for required probabilities p

S0

T0

STR decreases with time

MSÚFmax = Rmin

loading F,resistance R

reliabilityS [ % ]

t i m e 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 Ni cycles)

cut-off limit(permanent

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

Δσ

Wöhler’s curve

σ

1 cycle

stress 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

Δσ

aN = σΔlogloglog maN −

log Δσ

log N

N = 2 × 1 0

6

N = 5 × 1 0

6

N = 1 × 1 0

8

bilinear

trilinear

designationof category

ΔσC

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 (ΔσR) 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

Δσ

N

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 given in Eurocodes)




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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:

Δσ 1

Δσ 2 Δσ 3 Δσ 4

1 1

2

3

4

5

2

3

4

5

1

23

454'3'

history after

filtration

idea of "pagoda"

(turned of 90º)

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

N

Δσ n1 (for stress range Δσ1)n

2 n3 histogram:

ΔσΔσ1

Δσ2Δσ3

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 f y is negligible(steel S235 and S460 have roughly the same fatigue strength).

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

•;

+

-

Δσ

Δσ ( in compression may be taken 60% of6 Δσ only)

NOTCHES are concentrators of stresses→ cracks,

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




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

The design may be performed for:

• constant amplitude of nominal (equivalent) stress range ΔσE,2,• stress range spectrum.




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Design for „constant equivalent stress range amplitude"

For direct stresses:

Mf

CE,2Ff

ΔΔ

γ

σσ ≤

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

3Δσ

aN =

5

Δσ

aN =

const.

1 6 0

160

cut-off limit

cut-off limit for

„constant amplitude"117

2.106

5.106108 cycles N

Δσ R




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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.

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

1n

i Ri

EidNnD

number of cycles with amplitudeγFf Δσi

number of cycles with the sameamplitude up to collapse, determinedfrom curve corresponding tocategory of given detaillog NNRi

logΔσ

nEiFf Δσi




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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 (Δσ, 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

Ö KD 80




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Complementary notes:

Fracture mechanicsUnlike 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-cyclefatigue (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 → 02a

b

r

K

πσ I2

max =

coefficient of stress intensity (after Irwin).

Solution consists of:

b)(a,f aK πIa) Stress in crack face:

b) Velocity of crack spread (Paris’ law): m

d

dKC

N

aΔ

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 („fracture toughness – material constant)

a „critical length“ of the crack acr 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

ccr

1⎟⎠

⎞

⎜⎝

⎛=

f

Ka

σ

I

( )

∫cr

0 ΔK

da

a f

aN

log Δσ

log N

approx. 10 000 cycles

quasi-static fracture

low-cycle fatigue

multi-cycle fatiguecut-off limit

Nonlinear fracture mechanics (low-cycle fatigue)

Region of plastic deformations – use of Δε pl necessary

σ

ε

ε pl ε el

ε tot

Manson-Coffin relation:

Manson relation:

( )C, N2pl ε= 2N number of half-cycles

C constant (-0,5 up - 0,8)

ε ' 0,5 up 0,7εy

f y' coeff. of fatigue strength ≈ f yCb

pleltot )2()2)(( N'NE/f 'y ε+

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