(a) Type 1- If a bifurcation buckling analysis is performed using an elastic stress analysis without geometric nonlinearities in the solution to determine the pre-stress in the component, a minimum design factor of ΦB = 2/ ßcr shall be used (see 5.4.1.3). In this analysis, the pre-stress in the component is established based on Design Load Combinations (1) through (9) in Table 5.3.
(b) Type 2 – If a bifurcation buckling analysis is performed using an elastic-plastic stress analysis with the effects of non-linear geometry in the solution to determine the pre-stress in the component, a minimum design factor of ΦB = 1.667/ßcr shall be used (see 5.4.1.3). In this analysis, the pre-stress in the component is established based on Design Load Combinations (1) through (9) in Table 5.3.
(c) Type 3- If a collapse analysis is performed in accordance with 5.2.4, and imperfections are explicitly considered in the analysis model geometry, the design factor is accounted for in the factored load combinations in Table 5.5. It should be noted that a collapse analysis can be performed using elastic or plastic material behavior. If the structure remains elastic when subject to the applied loads, the elastic-plastic material model will provide the required elastic behavior, and the collapse load will be computed based on this behavior.
A Type 2 analysis is not expected to produce noticeably different results to a Type 1 analysis for external pressure on a dished head. This article investigates the difference between a Type 1 elastic bifurcation buckling analysis and a Type 3 elastic-plastic analysis with explicitly modeled imperfection. A 24” OD ASME F&D head is analyzed with 0.25” wall thickness, SA-516 70 material at 200°F.
Type 1 Analysis
100 psi external pressure is applied. The first mode shape eigenvalue is 38.307. For dished heads ßcr is 0.124, therefore a minimum eigenvalue of 16.129 is required. Dividing the calculated eigenvalue by the required value is 2.375 which would indicate that the head is acceptable for 237 psi.
Type 3 Analysis
First an elastic-plastic analysis using a true stress strain curve is performed with the true idealized shape of the head analyzed. A target pressure of 500 psi is applied, and buckling is experienced at 340 psi. Dividing this result by the ASME VIII-1 required design factor results in an MEAWP of 97 psi.
The key point of the Type 3 analysis is “imperfections are explicitly considered in the analysis model geometry.” To fully comply with the code requirement, the deformed result of the bifurcation buckling analysis is imported into the elastic-plastic analysis as the initial shape. The analysis is re-run and the onset of buckling now occurs at 266 psi. Dividing this by the ASME VIII-1 design factor results in an MEAWP of 76 psi. If the design factor on buckling is taken as 2.0 this results in an MEAWP of 133 psi.
Conclusion
An external pressure calculation is performed in accordance with UG-28 of ASME VIII-1 and results in an MEAWP of 135 psi. This lines up very closely with the Type 3 analysis using a design factor of 2.0.
The above results show that the Type 1 analysis produces unconservative results and caution should be used if selecting this method. There is ongoing discussion within the ASME code committee about re-working the buckling rules and possible removal of Type 1 and Type 2 analyses.