Evaluation of “Linearized” Stresses Without Linearization

ASME Code stress assessment of pressure vessels in the power generation industry is usually done by finite element analysis using one of the two approaches. In the first, “shell-element” approach, vessels are modeled out of shell elements; primary plus bending and primary plus secondary stresses are taken directly from the finite element analysis results and the alternating stresses are based on primary plus secondary stresses prorated by respective stress concentration factors. The strength of the “shell-element” approach is its simplicity; its weakness is problematic modeling of the stress concentration and some modeling difficulties (varying wall thickness, nozzle/vessel connectivity, pressure applied to the mid-surface instead of to the inner surface.) In the second, “solid-element” approach, vessels are modeled out of solid elements; “linearized” stresses can not be taken directly from the finite element analysis results, first they must be linearized, and only then, can be compared against their allowable counterparts; the alternating stresses can be based directly on the outer/inner-surface-node-stresses, provided that the mesh of the model is fine enough to account for the stress concentration effect. The strength of the “solid-element” approach is its high accuracy; its weakness is the time consuming, sometimes ambiguous, stress linearization process. This paper proposes a modification of the “solid-element” approach, in which the time consuming linearization process is replaced by a modification of the original model. To do so, a vessel must be modeled out of quadratic 20 node solid elements; the mesh density of the model (on its surface and through thickness) must be adequate for stress concentration representation and the mesh lines in the thickness direction must be more or less normal to the surfaces. The results from this original model can be taken directly for fatigue evaluation. To obtain the “linearized” stresses the original model must be slightly modified, specifically the number of elements through thickness must be reduced to one, and the reduced integration technique is recommended. For such a modified model, the nodal stresses are equivalent to the “linearized stresses” of the original model. The equivalence is discussed on a model of a circular nozzle attached to a cylindrical vessel. The vessel loads are pressure and thermal expansion.