Pressure Vessel Stresses

The pressure vessel codes (including the ASME code and others) were originally intended to partner manual or hand calculation methods from which discrete values of stress can be obtained. Difficulties can arise when attempting to use them in conjunction with numerical analysis that produces a continuously varying stress field. This article explains some important concepts behind the pressure vessel codes, as well as issues that arise in practice when using them in partnership with FEA (termed ‘design by analysis’.) To improve understanding, some of the definitions from the codes have been simplified.

Primary and Secondary Stresses

The pressure vessel codes define two important ‘classes’ of stress. A primary stress is related to mechanical loading directly and satisfies force and moment equilibrium. Primary stress that exceeds the yield stress by some margin will result in failure. By contrast, secondary stresses are those arising from geometric discontinuities or stress concentrations. For an increasing external load, at any point, both primary and secondary stresses increase in proportion to this load, until the yield point is reached. But secondary stresses are termed self-limiting by the ASME code: that is, once the yield point has been passed locally around the stress concentration, the direct relationship between load and stress is broken, due to the reduced post-yield stiffness of the material. This is in contrast to primaries (sometimes termed ‘load controlled’ stresses) that will continue to increase in overall magnitude, in direct proportion to the applied load, irrespective of the shape of the stress-strain curve, until failure.

In a region away from any discontinuities, only primary stress will arise. The secondary stress cannot arise alone however - at a discontinuity, the secondary stress will be superimposed on the underlying primary stress. It is worth pointing out the distinction made between primary and secondary stress in the pressure vessel codes is broadly similar to that made between net section and peak stresses identified in the British Standards for the assessment of fabricated structures, as described in previous articles.

Primary stresses are further categorised into a uniform (single value) membrane stress distribution across a cross-section; and a linearly varying, bending stress distribution. These definitions are more ambiguous in the codes than those for primary and secondary stress but are necessary since they have different allowable values.

Allowable Stresses Based on Failure Modes

Because different modes of failure are associated with primary membrane, primary bending and secondary stress, different allowable values are defined for each. These are not given as absolute values in the pressure vessel codes, but as a proportion of the yield stress of the material in question.

Primary membrane stresses are not allowed to exceed yield otherwise there is the possibility of a catastrophic plastic collapse e.g. a burst under pressure. For a membrane stress, the limiting value will be reached over the full vessel cross-section simultaneously and a margin of safety is included by specifying an allowable membrane stress of 2/3 yield. The total primary (membrane plus bending) allowable stress is greater, having a value of yield, because the bending element means that the stress will only be reached at a location in a localised crosssection, most distant from the neutral axis. Secondary stress can comfortably exceed yield but must be limited to ensure shakedown under cyclic load. Hence the range of secondary stress is limited to twice yield, as explained in the previous article. High cycle fatigue considerations are also addressed by the code but these are not discussed further here.

Identifying Primary and Secondary Stresses

The biggest problem when using continuum stress field results produced by FEA in conjunction with the codes is to decompose the total stress at a point into the primary and secondary stresses. In the long term, additional code rules may be developed that consider better how to do this. One solution the analyst has currently is to define hand calculations to relate the primary stress to the applied direct and bending loads. If this cannot be achieved, an analyst would not be able to show how much of the stress at a point is secondary and, for conservatism therefore, he would have to consider it all as primary. This results in the undesirable situation of comparing peak stress (which may be mainly secondary) with the primary allowable stress (2/3 yield) which can give very conservative designs.

Another alternative might be to select a cross-section away from stress concentrations but not from the load path, where primary stresses are at least the dominant part of the total stress distribution. However, in practice, it is difficult to define how to locate such a section and to do it to the satisfaction of clients who may be unconvinced by reasoning that ignores the maximum stress.

Many postprocessors have a facility to decompose the bending and membrane parts of primary stress across a cross-section in a vessel (called stress linearization). These procedures assume all stress is primary however and can also be over-conservative.

The Inelastic Approach

Differentiating between primary and secondary stress is an indirect method of accounting for the different failure modes of the vessel. Given the difficulties described and the power of modern analysis, it is not surprising that new approaches are being discussed. One such is to represent the failure modes directly in the FEA by an inelastic analysis. This will be discussed in subsequent articles.