Linearisation Guidelines

It has been apparent for some time that there are deficiencies in the rules for design by analysis when the finite element method is used. In particular this has highlighted problems with the design criteria and the underlying philosophy of assessment. Over the past few years, the US Pressure Vessel Research Council (PVRC) has funded a project to consider recommendations for updating the ASME Code. It is worthwhile reviewing some of these recommendations; a summary has been given by Hechmer & Hollinger[9].

The short term recommendations consisted of six sections. The second, fourth, fifth and sixth recommendations are related to linearisation problems for primary stress and three-dimensional problems. The first and third recommendations are of a more fundamental implication since they relate to the use of finite element methods for design by analysis using the existing ASME - Code criteria. The project members have been very careful with the wording of the recommendations, and some interpretation is required. These recommendations consider essential pressure vessel components, which are basic structural

elements:

• Shells of revolution and circular plates with either constant or variable thickness (transition elements) - normally connecting one structural element to another.

• Smooth junctions - where the model represents the actual geometry for example connecting fillet or blend radius.

• Sharp junctions - where the model does not represent actual geometry, such as sharp corners or notches, as shown in Figure 2.25. Basic Structural Element Sharp Junction Transition Element Smooth Junctions Fillet Blend Sharp Junction Basic Structural Element

Design by Analysis

2.30 Design by Analysis

The first and third recommendations are summarised below:

First recommendation: This relates to the use of finite element analysis (FEA) in pressure vessel design by analysis. It is recommended that for the majority of pressure vessel components, which are basic structural elements, FEA is inappropriate. Pm stresses should be calculated using general

equilibrium considerations, with Pm+Pb evaluated by hand calculations for conditions where Pm is

small (for example in flat plates). FEA is appropriate for calculating PL+Pb stresses near

discontinuities (see third recommendation below) and for the calculation of P+Q stresses in general. Notably it is only in complex components where basic structural analysis does not exist that FEA is recommended as appropriate for Pm and Pm+Pb stress evaluation. “... the thrust is that the designer

should be applying his ingenuity to calculating equilibrium stresses, not to extracting stresses from a general finite element model ...”.

Third recommendation: This relates to the locations in a pressure vessel where stress evaluations for Code compliance should be considered. It is recommended that it is appropriate to perform Pm+Pb

(PL+Pb) and P+Q evaluations in basic structural elements, but inappropriate in discontinuity type

transition regions. If there is a smooth junction then the stresses should be evaluated in the row of elements adjacent to the junction (or the line of nodes at the junction). When there is a sharp junction, the evaluation must be far enough from the junction so that the stresses are not affected by the notch behaviour. This recommendation should eliminate the need to linearise erratic stress distributions; “... the thrust ... is that plastic collapse and gross strain concentration will not occur in stiff transition regions; they will occur in the more flexible shell elements ... the purpose of the P+Q limits is to validate the fatigue analysis by precluding strain concentration and ratchet. It is highly unlikely that ratchet could occur in a transition element ...”

The first recommendation is rather subtle. In the light of the ASME Code (as it stands), finite element analysis is only appropriate in certain special cases in primary stress calculation - in general, equilibrium and shell discontinuity analysis are to be preferred. However, FEA is appropriate for secondary (and peak) stress evaluation. In the context of the discussion given this may be interpreted further as follows: finite element analysis may be used to evaluate the overall stress distribution for shakedown and fatigue assessment but the analyst should use simple calculations and strength of materials arguments to extract the primary stress components. In other words, elastic finite element analysis should not be used as the basis for categorisation or evaluation of primary stress.

The third recommendation also needs careful interpretation and is the most intriguing of all those provided by the PVRC project. The implication to the writers is clear - ignore the calculated stresses in sharp transition regions, since they will not affect the post yield failure mechanisms.

The mid term recommendations aim to provide additional tools and procedures to assist the designer in making better use of the existing ASME Code rules, specifically to address the problems of categorisation and linearisation directly through finite element analysis.

Finally the long-term recommendations aim for a more fundamental assessment of the ASME Code philosophy and criteria and require extensive new research. It is felt that new rules should be based on specific quantities required to prevent a failure mechanism, perhaps moving away from simple elastic analysis and stress evaluation. For example, the limits based on shell type membrane and

bending stress are difficult to understand and often misinterpreted, while the secondary limits are probably oversimplified and over-conservative, particularly in the presence of combined load. Considerable research on shakedown and ratchetting over the past twenty five years has confirmed this.