Issue 31

A.R. Maligno et alii, Frattura ed Integrità Strutturale, 31 (2015) 97-119; DOI: 10.3221/IGF-ESIS.31.08 103 a / B K I (Analytical solution) Bending stress 1 MPa K I (FE solution) Bending stress 1 MPa 0.2 2.999 2.981 0.4 4.401 4.364 0.6 5.585 5.522 0.8 6.703 6.592 Table 3 : Comparison of K I for analytical and numerical solutions. In this section, the initial crack size and material coefficients of the Paris crack-growth law were evaluated and calibrated against the Quality Category 2 (Q2) S-N curve as given in BS 7910 Code. In order to evaluate the residual life, the Paris equation was used:   m KA dN da   (5) where the coefficients A and m for the steel under analysis are: A = 5.21x10 -13 ; m = 3 for da/dN in mm/cycle and  K in MPa mm 1/2 . Minimum crack size The size of the elliptical initial crack on the inner wall of the pipe was also estimated. It was of paramount importance that the estimated number of cycles for each crack size, at different stress range levels, was reasonably close to the fatigue cycles of the Q2 S-N curve. In BS 7910 Code, the quality categories refer to particular fatigue design requirements or to the actual fatigue strength of flaw-containing components. The quality categories are defined in terms of the ten S - N curves labelled Q1 to Q10. These are described by the following equation [18]: 3    constant (6) The value of such constant for the Quality Category 2 (Q2) for steel is: 1.04×10 12 cycles. At a stress range of 80 MPa, the estimated life is 2×10 6 for X80 steel. This reference stress and the number of fatigue life cycles were used in the first iterative calculations. The numerical integration of the Paris law (Eq. 5) was carried out using a fixed increment da . After each increment of crack advance an average number of life cycles was determined as     1 1 1 2 / / i i i da da dN da dN da dN              (7) The total life is therefore: N i+1 = N i + dN i+1 (8) The SIFs were calculated using the fitted curve described by Eq. 1 for various a/B values. After a number of iterations the evaluated crack depth a that gave a residual life estimate of 2×10 6 cycles at the reference stress range of 80 MPa was 3.92 mm. A comparison of cycle estimations at various stress ranges, namely, 300 MPa, 80 MPa, 50 MPa, 20 MPa, and at different initial crack sizes a (3.81 mm, 7.62 mm, 3.92 mm) is shown in Fig. 7. The values 3.81 mm and 7.62 mm were chosen as they represent a/B =0.1 and 0.2. The initial crack size of 3.92 mm matches the S-N curve for all the stress ranges considered and, therefore, it was used for the preliminary finite-element analyses of crack growth. During this analytical study, the SIFs were constantly monitored and checked against the fracture toughness ( K IC ) value for X80 steel (225 MPa  m) to detect any failure due to fracture. The analytical solution showed that the stress intensity factor during the crack propagation did not exceed the K IC value; failure was achieved due to the break-through condition.