Issue 28

D. Gentile et alii, Frattura ed Integrità Strutturale, 28 (2014) 42-50; DOI: 10.3221/IGF-ESIS.28.05 45 T HE HODGRAPHCONEMETHOD he hodograph cone method (HCM) is a geometry approach for the estimation of the COD profile for a generic circumferential part-through crack in a pipe under bending and/or internal pressure. The method and the derivationof the solution is given in details elsewhere [1]. In the following theHCM is summarized. According to theHCM, the geometrical representation of the COD of a circumferential crack in a pipe can be seen as the intersection of two surfaces: a cylindrical surface, representative of the pipe, and a cone, with elliptical base and the vertex insisting on the cylinder circumference. The base of the cone is an ellipse with itsminor axis equal to themaximum crack opening displacement for the in axis configuration. The cone has an opening angle that is half of the physical crack opening angle. A sketchof theHCM is shown inFig. 3. Figure 3 : Geometric representationof the hodograph cone for the reference in-axis configuration. For the centered crack configuration, the intersectionof the cone and the cylinder return in the 3D space theCODprofile as the elliptical profile projected onto the pipe surface. For the off-centered crack, theCODprofile can still be obtained as the intersection of the cylinder and the shifted cone, moving its vertex along the cylinder circumference by the same off axis angle  , Fig. 4. For a circumferential crack length 2  R and a generic off-centered angle  , the COD curve in the 3D space can is given as:                     2 1 2 2 2 2 2 / * cos cos sin sin tan sin cos y x R z R                        (1) where   ;       and *  is the maximumCOD at the center crack length for the in axis configuration. Off course this value is function of the pipe geometry and load and can be determined either by FEM or estimated according to GE- EPRI or similar solutions. According to this, crack closure starswhen the crack goes in the compression region that is predicted tooccurs at an angle that depends on the crack length according to,   1 2      (2) T