Size the reinforcement rings of all your openings with 2pDeE. Envelope formula of mechanical strength through a new synthetic approach: Ordinal Correlation Algorithm

Openings (nozzles, branch connections) located in the knuckle region of pressure vessel heads represent a major design challenge, particularly when neighboring nozzles induce stress concentrations in the ligament separating them. Current construction codes, such as CODAP and ASME BPVC.VIII.1-2 , do not provide explicit rules for these configurations, often leading to lengthy and costly finite element (FE) analyses. The objective of this study is to develop a simple and parsimonious analytical envelope formula linking the maximum equivalent stress to geometric and mechanical parameters, in order to derive design rules (minimum reinforcement thickness, allowable pressure) ensuring an adequate level of safety. The methodology is based on a parametric FE database ( Abaqus ) covering several thousand configurations of neighboring and isolated nozzles. An original algorithm constructs an ordinal correlation matrix to identify critical zones and establish an objective hierarchy of local stresses. An incremental weighting, called ordinal regularization , applied to the ratio D e / E , leads to an explicit envelope formula for the Tresca equivalent stress, expressed as 2 p D e E , used for the sizing of the reinforcement ring: E ring ≥ 2 p D e σ adm − E head .