Commentary On The Alternative Rules For Determining Allowable Compressive Stresses For Cylinders, Cones, Spheres And Formed Heads For Section VIII, Divisions 1 And 2
This Bulletin includes a commentary on the buckling rules proposed in WRC Bulletin 406, as an alternative to the present buckling rules of the ASME Code. The proposed rules were used to develop the rules of ASME Code Case 2286 and are intended to be eventually incorporated in the ASME Code. This commentary provides the basis for the rules proposed in Bulletin 406. It also points out the limitations of these rules. Comparisons are provided between the proposed rules and test results. The basis for the interaction rules of the alternative rules are also provided and compared with failure theories.
Proposed Rules for Determining Allowable Compressive Stresses for Cylinders, Cones, Spheres and Formed Heads
The purpose of this report is to propose alternative rules to those given in UG-23(b), UG-28, UG-29, UG-33 and Appendix 1-8 of ASME (1992a) for determining allowable compressive stresses for unstiffened and ring stiffened circular cylinders and cones and unstiffened spherical, ellipsoidal and torispherical heads. The allowable stress equations are based upon theoretical buckling equations which have been reduced by knockdown factors and by plasticity reduction factors which were determined from tests on fabricated shells. The research leading to the development of the allowable stress equations is given in API 2U (1987), Miller (1991), Miller and Saliklis (1993) and the Commentary.
This report expands the coverage of load conditions and shell geometries and includes equations for combinations of loads which are not given in ASME (1992a). The proposed rules also apply to shells with higher D/t ratios. Allowable compressive stress equations are presented for cylinders and cones subjected to uniform axial compression, to bending moment applied over the entire cross-section, to external pressure, to loads which produce in-plane shear stresses and to combinations of these loads. Allowable compressive stress equations are presented for formed heads which are subjected to loads which produce unequal biaxial stresses as well as equal biaxial stresses. More accurately equations are given for determining the size of stiffening rings.
Development of Design Rules for Conical Transitions in Pressure Vessels for the ASME B&PV Code, Section VIII, Division 2
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; and
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.