Advances in COD Equations — Circumferential Through-Wall Cracks
Authors: Young Bruce A. , Olson Rick J. , Kerr Matthew 
Published: August 2013

Non-linear fracture mechanics equations for through-wall cracks in a pipe are used to analyze piping systems for either critical flaw size or critical loading conditions as part of probabilistic Leak-Before-Break (LBB) failure analyses under the eXtremely Low Probability of Rupture (xLPR) program co-sponsored by the U.S. Nuclear Regulatory Commission (US NRC) and the Electric Power Research Institute (EPRI). The xLPR analysis techniques use a large number of independent analysis solutions to determine an overall assessment of system failure probability. As part of the assessment, each independent solution requires the solution of the crack opening displacement (COD) for a through-wall crack (TWC) in a pipe under the prescribed loading conditions. The COD evaluations are then used to determine a leak rate for the given load conditions and crack sizes.
The purpose of this paper is to present results which advance the start-of-the-art for determining the elastic-plastic functions for crack opening displacements (COD) for a TWC in a pipe system under combined tension and bending loads. The current method used to determine COD in xLPR, a blending of tension and bending solution from the GE-EPRI Handbook, determined the continuum equations using structural finite element analyses with shell type elements. Since that body of work was undertaken, there have been significant advancements in computing capability such that structural finite element analyses with three-dimension continuum elements are currently feasible. The use of continuum elements provides several advantages over shell elements; such as, the ability to elicit details of variation in the COD through the thickness of the pipe wall and to apply pressure to the crack face due to the internal pipe pressure. Furthermore, the original GE-EPRI solutions were limited for the case of combined tension and bending loads. The existing GE-EPRI solutions for combined loading conditions are limited to pipe radius-to-wall thickness (R/t) ratios of 10 or greater, typical of those piping systems found in the boiling water reactor (BWR) fleet. For the PWR piping systems of concern today, which are subject to primary water stress corrosion cracking (PWSCC), the R/t ratios are typically 5 or less.
As a result of the limitations with the existing GE-EPRI method for predicting COD, Battelle and US NRC staff set out to develop a comprehensive COD prediction tool for combined loadings which would be applicable to both PWR as well as BWR piping. This effort involved a matrix of over 1,200 finite element analyses for a full range of pipe sizes, R/t ratios, through-wall crack (TWC) lengths, and internal pipe pressures. It is anticipated that there will be several parts to this effort. Part I, discussed in this paper, focuses on the development of the model and the initial investigation into the elastic- and elastic-plastic fitting functions for the prediction of COD (i.e., the V and h functions). Future parts of this effort will focus on such issues as the effect of restraint of pressure induced bending on COD, the effect of weld residual stresses on COD, J-Integral estimation schemes, and development of variable crack-face pressure.