FEM Calculations of SIF and COA for Through Axial Cracks in Pipes in Geometrically Nonlinear Formulation

The determination of stress intensity factors (SIF) and crack opening area or displacements (COA or COD) is important constituent when performing the “leak before break” analysis of piping systems in NPPs. The tabulated parametrical results of their calculation are widely presented in modern scientific and normative literature. Nevertheless, there is one aspect of crack behavior, at least in thin walled pipes, which still had not obtained its due attention. We mean here the geometrically nonlinear effect, which can be the big enough to be accounted for in practical applications. It is considered in geometrically linear analysis that only the inner pressure opens the crack, and COA and SIF are directly proportional to it. SIF is presented usually as solution for infinite plate multiplied by so-called bulging factor, BF, which depends on dimensionless crack length, i.e. ratio of crack length divided on square root of product of radius, R, and wall thickness, t. Two loading factors in thin walled pipes can contribute to geometrically nonlinear behavior. The first one is axial stresses induced by value of axial force or bending moment. The second one – is the inner pressure itself. The most attention in present paper is given to influence of the axial force. With this goal the numerical models were created for pipes with different ratios of R/t (20, 30, 40, 50) and different dimensionless crack length (2, 4, 6, 8). To exclude the nonlinearity due to circumferential stress the inner pressure is kept as a very small value and dimensionless SIF and COD values are calculated with respect to axial force. To prove the correctness of choosing the finite element types, meshing, number of elements along the thickness, loading steps the auxiliary problem of nonlinear modeling of transverse beam loaded additionally by very big axial force is considered. The very good correspondence was attained. For the pipe with axial crack the careful verification of numerical model was performed by comparison with linear results existing in literature. The results obtained are presented as a percentage of difference between the linear and nonlinear results. They show that influence of geometrical nonlinearity is fairly essential to be accounted in practice and can reach for practically real cases almost 3–10%. The change of SIF in percentages due to geometrical nonlinearity for different axial stress levels and for different crack lengths can be fairly well presented as unique dependence from product of stresses, radius to thickness ratio, and square root of dimensionless crack length. The change COD in central point of crack is slightly bigger than for SIF and the same unique dependence can be formulated for COD with only exception for small cracks λ < 3.