Pressure Vessel Design Review: ASME VIII DIV.1 and DIV.2 External Pressure Formulas

Verification under external pressure is one of the most critical aspects of pressure vessel design due to the risk of elastic or inelastic buckling. The ASME Boiler and Pressure Vessel Code (BPVC) provides two primary divisions for vessel design: Division 1 and Division 2. While both ensure safety, they diverge significantly in approach, methodology, and conservatism, especially under external pressure. This article explores the historical context, technical methods, and philosophical differences between the two divisions, focusing specifically on design by formula.

1. The ASME Section VIII Division 1 Approach to External Pressure

1.1 Empirical Origins: The Division 1 method for external pressure is rooted in empirical data combined with classic elastic stability theory. In the 1930s to 1950s, organizations such as the U.S. Experimental Model Basin conducted thousands of full-scale tests on cylindrical shells subjected to external pressure. Led by researchers such as Windenburg and Trilling the tests were validated on cylinders of various materials measuring collapse pressures and investigating failure modes across a wide range of geometries. The formulas proposed by Windernburg and Trilling are independent of the number of buckling ridges, which was the main difference from equations developed by other well recognized peers, such as Richard Von-Mises, whose theoretical shell buckling equations under external pressure are used in all European Pressure Vessel Codes.

The results were compiled into non-dimensional curves that relate geometry ratios (such as L/D and D/t) to collapse pressures. These curves were then processed to include conservative safety factors and adjusted to account for practical imperfections, such as fabrication-induced ovality and residual stresses. The final product was a set of standardized charts – “Chart A” and “Chart B” – now found in ASME Section II, Part D.

1.2 Use of A and B Charts: The allowable external pressure in Division 1 is determined using a two-step graphical process (simplified code provided herein):

• Determine shell geometry: outside diameter (Do), thickness (t), and unsupported length (L).

• Calculate the geometry ratios: Do/t and L/Do.

• Use these ratios to locate the corresponding “A” and “B” values on the ASME charts (e.g., Curve B).

• Calculate the allowable pressure (Pa) using formula: Pa = 4B/(3Do/t)

These charts embed large safety margins implicitly. There is no explicit distinction between elastic and inelastic buckling – both are lumped together through the conservative nature of the data.

2. The ASME Section VIII Division 2 Approach (Design by Formula)

2.1 Newer Theoretical Foundations: Division 2 uses formulas informed by the elastic theory first developped by Euler, Timoshenko, Bryan-Bress. The basis of these formulas lies in the work of Miller presented in the 1980s and 90s. Miller’s method was first adopted in Code Case 2286 and it is worthwhile mentioning that it is independent of the number of buckling ridges, just as the Windenburg and Trilling model. Miller’s method is also validated by extensive testing on carbon steel vessels.

To bring this theoretical formula into practical use, Division 2 applies reduction factors for imperfections and explicit safety factors. The inelastic buckling regime is addressed via separate formulas incorporating material yield strength and plasticity effects. The code provides transition criteria between elastic and inelastic design based on slenderness ratios and elastic critical pressure versus yield stress.

Conclusion

ASME Division 1’s external pressure design by formula is based on empirical charts developed from real-world collapse tests, offering simplicity and high conservatism. Division 2’s design-by-formula method is rooted in classical buckling theory, adjusted for real-world conditions. Each approach serves different engineering contexts, and understanding their origins and limitations helps engineers apply them appropriately during the pressure vessel analysis.