PSI - Issue 13

Konstantinos Kouzoumis et al. / Procedia Structural Integrity 13 (2018) 868–876 K. Kouzoumis et al. / Structural Integrity Procedia 00 (2018) 000–000

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2. Assessment of pipes with axial flaws

2.1. BS 7910

2.1.1. Through Thickness Flaws The current version of BS 7910, BSI (2013), provides an equation (P.17) for the calculation of reference stress in pipes with through thickness axial flaws under membrane stress and / or through wall-bending. Disregarding through wall bending, which can be ignored in most cases of pressurized pipes, it is expressed as, σ re f = 1 . 2 M T P m (1) Where M T is the Folias factor defined as, M T = 1 + 1 . 6 c 2 r i B (2) Eq. (1) includes two multiplication factors for translating the applied membrane / hoop stress ( P m ) to reference stress. The second multiplication factor on the applied stress is M T , called the Folias factor or bulging factor. Folias (1964, 1965, 1973) focused on solving the coupled di ff erential equations of the displacement, at the direction normal to the curved plane of the plate, and stress functions, with boundary conditions that take into account both the existence of the flaw and the curvature included in the geometry of a curved shell, in particular spheres and cylinders. The equations used assumed homogeneous, thin segments 1 of elastic shells subjected to small deformations and strains, so that the stress strain relationship is established by Hooke’s law. With the use of a fracture criterion based on Gri ffi th’s theory, Folias concluded a relationship between the failure loads for flat plate and curved shell, which was later invoked in the BS 7910 reference stress and stress intensity factor solutions. More specifically, after a series of deductions and the assumption that bending loads were of zero magnitude, Folias concluded the following equation for an axial cracked cylindrical panel (Eq.44 in Folias (1984)): (1 + 0 . 49 λ 2 )( ¯ σ e /σ ∗ ) 2 = 1 (3) Where the shell factor / parameter ( λ ) is, and the failure stress based on Gri ffi ths Theory σ ∗ = 16 G γ ∗ /π c (5) By setting the Poisson ratio ( ν ) as 0.3, interpreting ¯ σ e as the applied stress and σ ∗ as the reference stress, Eq. (3) becomes, σ ∗ = 1 + 1 . 61 c 2 r m B ¯ σ e (6) Eq. (6) is now in the same form as Eq. (2) with the only di ff erence that the mid-thickness radius instead of the internal radius is used. The remaining 1.2 multiplication factor, is justified in BS 7910:2013 BSI (2013), as “an empirical multiplication factor. . . based on experimental experience gained from large-scale pipe tests”. However, a di ff erent explanation was given in earlier revisions of the standard (BSI (1999, 2005)), namely; “The multiplier of 1.2 is introduced to achieve approximately similar levels of conservatism as that in P.3.1” (a reference to the solution for flat plates with through thickness flaws). This latter explanation does not seem very convincing, since the reference stress solution for a flat plate with a through thickness flaw is a simple loss of area correction. λ = [12(1 − ν 2 )] 0 . 25 c √ r m B (4)

1 The thinness of the shells reduced the problem to the two-dimensional linear theory; shells are considered thin if the radius of curvature (r) is greater by at least two orders of magnitude than the thickness (B), i.e. B / r ≤ 0 . 01