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**Published:** Jan 1976

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**Source: **STP590-EB

A two-dimensional finite element method is used to develop stress intensity factor solutions for continuous surface flaws in structures subjected to an arbitrary loading. The arbitrary loading produces a stress profile σ acting perpendicularly to a given section *S* of the structure. The stress profile is represented by a third degree polynomial

Stress intensity factor solutions are developed for continuous surface flaws introduced in particular sections S in the structure considered. Solutions are developed for a surface flaw in a flat plate, for both circumferential and longitudinal flaws inside a cylindrical vessel, and for circumferential flaws at several locations inside a reactor vessel nozzle.

The superposition principle is used, and the crack surface is subjected successively to uniform (σ = *A*^{0}), linear (σ = *A*^{1}*x*), quadratic (σ = *A*^{2}*x*^{2}), and cubic (σ = *A*^{3}*x*^{3}) stress profiles. The corresponding stress intensity factors (*KI(0)*, *KI(1)*, *KI(2)*, *KI(3)*) are then derived for various crack depths using the calculated stress profile in the region of the crack tip. The total stress intensity factor corresponding to the cracked structure subjected to the arbitrary stress profile is expressed as the sum of the partial stress intensity factors for each type of loading. *a* is the crack depth and *F*^{1}, *F*^{2}, *F*^{3}, and *F*^{4} are the magnification factors relative to the geometry considered. The results are presented in terms of magnification factors versus fractional distance through the wall (*a/t*) and reveal the strong influence of the geometry of the structure and of the crack orientation.

The stress intensity factor solutions obtained using this method are compared to solutions obtained using other methods, when available. In the case of the plate geometry, the solution obtained for the linear loading (σ = *A*^{0} + *A*^{1}*x*) is shown to agree well with the boundary collocation solution reported by Brown and Srawley. The stress intensity factor solutions for the circumferential and longitudinal cracks in the cylindrical vessel compare well with solutions obtained by Labbeins et al using the weight functions method proposed by Bueckner, and are also in good agreement with the solution for uniform loading (σ = *A*^{0}) obtained using the line spring method proposed by Rice.

**Keywords:**

crack propagation, nuclear reactors, pressure vessels, fracture properties, mechanical properties, stresses, deformation

**Author Information:**

Buchalet, CB *Senior engineers, Westinghouse Nuclear Energy Systems, Pittsburgh, Pa.*

Bamford, WH *Senior engineers, Westinghouse Nuclear Energy Systems, Pittsburgh, Pa.*

**Committee/Subcommittee:** E08.06

**DOI:** 10.1520/STP33958S