4 In the R6 Option 2 J estimation curve, the plasticity effect in the contained yielding regime is conservatively estimated by the simple term [5]: ref r ref E L ε σ2 2 1 Ω= (11) The constant of 2 in eqn. (11) comes from the plane stress condition, which provides conservative J estimates. Noting that pipes with circumferential part-through defects can be in between plane stress and plane strain conditions, the plane strain constant of 6, instead of 2, can be used for conservatism in COD estimation. However the value of 6 is valid only when the plasticity is confined within a dominantly elastically behaving body. The plasticity effect can be much more significant even below the widespread plasticity level. Moreover, the COD experiences a much enhanced plasticity effect, due to its local nature, as discussed in the above. Therefore, the plane strain assumption can be unduly conservative and eqn. (11) is retained for the proposed COD estimation equation, see the first equation in eqn. (1). In the literature, two models have been proposed to estimate the influence of plastic deformation before widespread plasticity, the Irwin-type correction model due to Kastner et al. [10], and the Dugdale-type correction model due to Wüthrich [11]. The proposed equation, eqn. (11), agrees well with both the Irwin and Dugdale-type correction models for typical materials. The proposed plasticity correction, eqn. (11), is also supported by experimental pipe test data, as will be shown in the next section. A final note is that continuously hardening materials show a smooth transition from elastic to plastic hardening behaviour. For such materials, the plastic behaviour within the 0.2% plastic strain also contributes (often significantly) to the plasticity effect in the contained yielding regime. EXPERIMENTAL VALIDATION OF COD ESTIMATION EQUATIONS The pipe test data are taken from two sources [3, 12], and are summarised in Table 1. The calculated results are compared with pipe test data in Figs. 2 to 8, each of which shows COD as a function of applied moment M, axial load N, or pressure P. In each case, two predictions are shown. One prediction uses full stress strain data, based on eqn. (1), which is termed the reference Option 2 curve (“Ref. Opt 2”). The other prediction is according to eqn. (7), assuming that only yield and tensile strengths are available, and is termed the reference Option 1 curve (“Ref. Opt. 1”). Some figures also include the predictions according to the GE/EPRI approach, which are simply reproduced from [3]. Note that the GE/EPRI results are based on the best Ramberg-Osgood fit based on full stress strain data. The results in Figs. 2-8 firstly show the expected order for the two different levels of present predictions: the prediction based on limited tensile data (Ref. Opt 1) is always more conservative for LbB arguments, than that based on full stress strain data (Ref. Opt 2). It can be clearly seen that the present predictions are much better than or at least similar to those based on the GE/EPRI method, and are overall in excellent agreement. CONCLUSIONS Modification to the reference stress approximation for the COD are reported in this paper, for circumferential through-wall cracked pipes under tension and under bending. Three modifications include: • re-defining the reference stress using an optimised reference load to minimise the dependence of the strain hardening and the geometry; • simplification of the small-scale plasticity term; and • use of a power law fit to stress strain data, instead of the actual data. A lower bound COD estimation equation is also given, similar to the R6 Option 1 J estimation curve, which is suitable when only limited tensile properties are available. The resulting estimation equations are simple
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