PSI - Issue 39

Dmitry O. Reznikov / Procedia Structural Integrity 39 (2022) 256–265 ReznikovD.O./ Structural Integrity Procedia 00 (2019) 000–000

262

7

2 ln / { } k N S  .

(17)

Nk  

In this case, the mean and the standard deviation of the extreme value distribution can be expressed in terms of v Nk and β Nk : max { } / Nk Nk E u      , (18)



(19)

max { }

S

 

,

6 Nk 

where 0,577   is Euler's constant. Moreover, it can be shown (Makhutov, Reznikov, 2014) that for N k >10 the asymptotic form according to expressions (14) and (15) turns out to be close to the exact solutions according to expressions (12) and (13) and can be used in engineering calculations. Thus, the distribution of the random parameter Σ max can be expressed in terms of the moments of distribution of the parameter Σ i . 4. Numerical procedure for failure probability assessment Consider a pipeline component with an axial crack on the inner surface (Fig.1), loaded by an internal pressure that varies cyclically with constant amplitudes, on which a multiple tensile overloads are superimposed. We will assume that: (1)the parameters a 0 , C and m of the kinetic equation (6) are probabilistic and distributed according to the uniform density law ( a 0 ~ U (α a o ;β a o ), C ~ U (α C ;β C ), и m ~ U (α m ;β m ) , where α x and β x are, respectively, the left and right boundaries of the interval of values of x with a nonzero probability density, and (2) the intensity of the peak values of nominal hoop stresses at overloads are distributed according to the normal density law N ( E {Σ}; S {Σ}). The kinetics of a surface crack in the pipeline component is described by the block representation of the operational loading (Fig. 3) using equations (3) and (6). The condition of the component fracture is written in the form (11). The values of parameters of the numerical example are given in Table 1. Table 1.Numerical parameter values. Parameter Distribution / Value

Parameter type (deterministic / probabilistic) Deterministic Deterministic

Diameter of the middle surface of the pipe D , m

1.26

Thickness of the wall δ , m

0.043

The average value of the internal pressure in the constant amplitude loading cycle , p 0 , MPa Stress ratio in the constant amplitude loading cycle, R r Maximum hoop stress in the constant amplitude loading cycle , S max , MPa

Deterministic Deterministic

8.0 0,9

Deterministic Deterministic Deterministic Probabilistic Deterministic Probabilistic Probabilistic Probabilistic Deterministic

117.2

10,000

Number of the constant amplitude cycles, N r

1.85

Overload factor, k OL

Peak of the hoop stress in the overload cycle Σ i , MPa Mean of stress ratio in the overload cycles E { R Σ }

Normal N (216.8; 21.6)

0.45

Initial crack depth, a 0 , m

Uniform U (1.00∙10

-2 ;1.20∙10 -2 )

Parameter C of the Paris equation, m/cycle (Pa√m) -m

Uniform U (1.0∙10 -11 ;3.0∙10 -11 )

Parameter m of the Paris equation Fracture toughness K Ic , МPа√м

Uniform U (2.8; 2.85)

61.0