PSI - Issue 38

Luca Vecchiato et al. / Procedia Structural Integrity 38 (2022) 418–427 L. Vecchiato et al./ Structural Integrity Procedia 00 (2021) 000–000

422

5

(

)

; (

)

; (

)

(

) 2 h

(

)

(

) 2 i

2

2 w2 = ⋅ f

2 w1 = ⋅ f

2 w3 = ⋅ f

∆σ

∆τ

∆σ

∆σ

∆σ

∆τ

(7)

eq,peak,II

r , 0,peak θ θ=

eq,peak,I

, 0,peak

eq,peak,III

z, 0,peak

θθ θ=

θ θ=

j

j

i

h

Then, the Palmgren-Miner linear-damage rule must be applied, separately for each loading mode (see Fig. 2a,b,c), to derive an equivalent peak stress which generates the same damage of the n load levels (i.e. n σI , n τII and n τIII ) previously defined, but by acting for n 0 cycles, i.e. the same number of cycles for all loading modes. To do so, the PSM master curve valid for the considered loading mode is adopted according to Table 2, e.g. when dealing with steel welded joints under mode I loading (λ = 0) , the inverse slope is k 1 = 3, while under mode II and mode III (λ → ∞ ) the inverse slopes are k 2 = k 3 = 5 (Table 2): ( ) ( ) I 1 1 1 n k k I i eq,peak,I eq,peak,I i i 1 0 n n σ =   ∆σ = ∆σ     ∑ ; ( ) ( ) 2 II 2 1 n k k II j eq,peak,II eq,peak,II j j 1 0 n n τ =   ∆σ = ∆σ       ∑ ; ( ) ( ) III 3 3 1 n k k III h eq,peak,III eq,peak,III h h 1 0 n n τ =   ∆σ = ∆σ     ∑ (8)

(b)

Δσ eq,peak,II

Δσ eq,peak,I

(n I ) tot is the number of cycles of mode I which occurs in the reference duration of the analysed load history (a)

(n II ) tot is the number of cycles of mode II which occurs in the reference duration of the analysed load history

k 1 =3

(Δσ eq,peak,I ) 1

(Δσ eq,peak,II ) 1

k 2 =5

(Δσ eq,peak,II ) 2

(Δσ eq,peak,I ) 2

Δσ eq,peak,I

…

Δσ eq,peak,II

Δσ eq,peak,II,NA

Δσ eq,peak,I,NA

…

(Δσ eq,peak,II ) j

(Δσ eq,peak,I ) i

…

…

1

n σI

2

1

2 …

j

n τII

…

i

n

n 0

0

N A

N A

…

N

N

…

(n I ) 1 (n I ) 2

(n I ) i

…

(n II ) 1

(n II ) 2

…

(n II ) j

(n I ) tot

(n II ) tot

Δσ eq,peak

(c)

(d)

Δσ eq,peak,III

(n III ) tot is the number of cycles of mode III which occurs in the reference duration of the analysed load history

(Δσ eq,peak,III )

k=f(λ)

k 3 =5

(Δσ eq,peak,III )

Δσ eq,peak

Δσ eq,peak,III

Δσ eq,peak,III,NA

(Δσ eq,peak,III )

Δσ eq,peak,N A =f(λ)

…

n τIII

1

2

h

n … 0

N A

N

N A

N

N

…

…

…

(n III ) 1 (n III ) 2

(n III ) h

(n III ) tot

Fig. 2. Palmgren-Miner linear-damage rule applied to derive an equivalent peak stress which generates the same damage of the n load levels (i.e. n σI , n τII and n τIII ), but by acting for n 0 cycles for: (a) mode I, (b) mode II and (c) mode III contributions. (d) Overall equivalent peak stress compared with the proper design curve to estimate the fatigue life of the considered welded joint under VA loading conditions.