Development of Design Rules for Conical Transitions in Pressure Vessels for the ASME B&PV Code, Section VIII, Division 2

by David A. Osage, P.E., ASME Fellow / James C. Sowinski, P.E. / January 2010


The technical background behind the new ASME B&PV Code Section VIII Division 2 rules, for the design of conical transitions with and without a knuckle or flare are presented in this publication. The ASME B&PV Code Section VIII Division 2, 2006 Addenda and earlier, had a design-by-rule procedure in place to evaluate the adequacy of the cylindrical-to-conical shell transitions without knuckles or flares, for both internal and external pressure. For the internal pressure case, the inclusion of supplemental loads due to axial forces and moments are not permitted and the protection against plastic collapse is addressed through a figure look-up of an adequate thickness envelop. The thickness of the cylinder and cone at the junction did not need to be increased beyond that required for internal pressure loading based on the location of an assessment point on a plot with the cone half apex angle as the abscissa and the pressure-to-allowable stress ratio as the ordinate. If the assessment point falls above the adequate thickness envelop, additional thickness is required and the shell and cone shall be locally increased. The use of external reinforcement from a structural section is prohibited. For the external pressure case, supplemental loads due to axial forces and moments are permitted, and the protection against collapse from buckling is addressed through the calculation of a required area of reinforcement and compared to an available area either integral, external, or a combination of the two. Further calculations are required to determine the required moment of inertia of the cylinder-to-cone junction, when the junction is referenced to be a line of support, and compared to the available moment of inertia of the junction. This calculation procedure is identical to the one provided in ASME B&PV Code Section VIII Division 1, Appendix 1-8.


Due to inconsistencies between two methods regarding calculation procedures and the consideration of supplemental loads (i.e., supplemental loads are not included for internal pressure design and supplemental loads are included for external pressure design), and the lack of a step-by-step design procedures, a new consistent approach to evaluate cylindrical-to-conical shell transitions is developed.


The new design procedures for cylindrical-to-conical shell transitions, with and without knuckles or flares, were developed using two different approaches. However, common requirements for both design procedures included:


  • applicable loadings were to include pressure, axial forces, and net-section bending moments
  • reinforcement area must be integral
  • applicable to cone half apex angles ranging from 0 to 60 degrees
  • determination of the stress state developed at the cylinder-to-cone/knuckle/flare junction
  • a concise self-contained procedural layout
  • validation using finite element analysis


For cylinder-to-cone transitions without knuckles or flares, the design rules were developed based on thin shell theory and are an extension of the rules in ASME B&PV Code Case 2150. A matrix of cylinder-to-cone geometries was developed and stress analysis was performed considering the applicable loadings. The resultant shear and meridional bending moments acting at the junction were determined for each of the geometries and loadings, and curve-fit equations were developed to approximate these variables. Parametric equations were developed in close form which used the approximated resultant forces and moments to calculate the resulting membrane and bending stresses developed at the cylindrical-to-conical shell junctions for both the large end and small end junctions.


The design rules for conical transitions with knuckles or flares were formulated based on the pressure-area method. For axisymmetric shells, the pressure-area method requires the pressure forces which act normal to a plane containing the axis of revolution and act over a pressure area to be balanced by the circumferential or normal forces which act on the arc of the shell adjacent to the pressure area. Application of the pressure-area method to the knuckle and flare permit the calculation of the circumferential membrane stress. The longitudinal membrane stress was derived using membrane thin shell theory.


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