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

Transcription

1 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. reliability S [ % ] S 0 time T S T R decreases with time T 0 MSÚ F max = R min loading F, resistance R 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 OK3 1

2 Fatigue tests (see mechanical tests of material, bachelor course) σ stress range + σ regimes: pulsating tension 1 cycle N cycles (time) alternating loading pulsating compression Wöhler s curve low-cycle fatigue (< about cycles, plastic behaviour) multi-cycle fatigue (elastic behaviour) time strength" (for N i cycles) design fatigue strength curve e.g. for surviving with probability of p = 95 % (hyperbola) cut-off limit (permanent fatigue strength) N i [ N ] number of cycles up to damage OK3 2

3 Wöhler s curve in log coordinates (S-N curves, stress-number of cycles curves): log designation of category C bilinear trilinear Usually expressed in the form: N = a m i.e. logn = loga mlog N = N = N = log N 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 ( R ) is given especially by time of crack propagation up to critical length (fatigue fracture). OK3 3

4 Determination of loading effects Actual loading has stochastic distribution. σ Dynamic effects are taken into account: T - by dynamic calculations, - approximately with help of dynamic coefficient ϕ fat (given in standards). In fatigue design may be used: 1. Constant amplitude of stress range N and N are approximately estimated. In Eurocodes is determined equivalent stress range E,2, which corresponds to fatigue damage of N = : E,2 = λ 1 λ 2 λ 3... k product of equivalent damage factors (for bridges and cranes given in Eurocodes) OK3 4

5 2. Stress range spectrum Actual stress distribution is evaluated by some of the cycle counting methods, e.g.: - reservoir method: rainflow method: history 2 4 after filtration idea of "pagoda" (turned of 90º) ' 4' 2 4 The stress ranges are arranged into several degree spectrum (for several ): n 1 (for stress range 1 ) n 2 n 3 histogram: N n 1 n 2 n 3 n 4 OK3 5 N

6 Determination of fatigue strength Influence of stress range. is substantial. The fatigue strength in compression is higher: + - ( in compression may be taken 60% of6 only) Influence of stress concentration is essential: Influence of yield point f y is negligible ATTENTION: welded elements have always tension residual stresses in weld location!!! always tension. NOTCHES are concentrators of stresses cracks, they are especially at weld locations (see detail categories). (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. ; OK3 6

7 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 ) 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 E,2, stress range spectrum. OK3 7

8 Design for constant equivalent stress range amplitude" For direct stresses: (similarly for shear) γ Ff E,2 γ Mf C fatigue strength" for cycles given by name of detail category stress range of equivalent nominal stress (must be < 1,5 f y, including dynamic coefficient ϕ fat ) Detail categories R N = a 3 const. Curves in log scale: DC 36, 40, , 160 cut-off limit for constant amplitude" cut-off limit N = a cycles N OK3 8

9 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 ( i, n i, see e.g. for i = 4) the Palmgren-Miner linear damage accumulation hypothesis may be used: log D d = n nei i N Ri number of cycles with amplitude γ Ff i 1 γ Ff i n Ei N Ri log N number of cycles with the same amplitude up to collapse, determined from curve corresponding to category of given detail OK3 9

10 Example of crane girder: max. 100 KD 80 older opinions, today frequently welded KD 80 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 (, 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). t KD 80 manual weld: KD 100 MAG, SAW: KD 112 OK3 10

11 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 2a b r 0 σ max = 2K I π r coefficient of stress intensity (after Irwin). K I may be determined by FEM. Solution consists of: a) Stress in crack face: b) Velocity of crack spread (Paris law): K I = σ da dn N number of cycles C, m material constants ΔK amplitude K i.e. (K - K min)/2 I I,max I, π a f (a,b) = C ΔK m OK3 11

12 For given K I = K IC ( fracture toughness material constant) a critical length of the crack a cr may be determined: a and by integration of Paris law also the residual life (i.e. the number of cycles up to fatigue damage): cr N 1 K = π σ f a = cr da f ( ) a0 ΔK Ic (acr,b) 2 Nonlinear fracture mechanics (low-cycle fatigue) log quasi-static fracture low-cycle fatigue multi-cycle fatigue cut-off limit Region of plastic deformations use of Δε pl necessary. Energy of deformation is determined by J integral. σ ε approx cycles log N ε pl ε el ε tot Manson-Coffin relation: Manson relation: ( 2 ) C, N Δ ε pl = ε 2N number of half-cycles b C tot = Δε el + Δε pl = ( f ' y / E)(2N) ε' (2N) Δ ε + C constant (-0,5 up - 0,8) ε' 0,5 up 0,7ε y f y ' coeff. of fatigue strength f y OK3 12