Protection Against Plastic Collapse: Which Analysis Method to Use
The Design By Analysis Requirements, Part 5 of Section VIII Division 2 of the ASME Boiler & Pressure Vessel Code, gives three acceptable analysis methods for ensuring the design of a pressure vessel is adequate against failure due to plastic collapse. These methods are: elastic stress analysis method, limit-load analysis method, and elastic-plastic stress analysis method.
The details of the assessment procedures and acceptance criteria of these three methods are quite different such that the decision of which analysis method to use is not always easy. A particular requirement of one method can be less conservative than another method making it a more attractive option, while the opposite is true for another requirement of the same method.
The ASME Code does not provide any recommendations on which method to use because of the variability in approaches and design processes. Agreeing with this position, the objective of this article is not to opine on which method is the best but rather provide commentary on some of the requirements of the various methods that will hopefully assist the engineer with their decision.
Be aware that this article does not discuss all of the requirements, details, and limitations of the various methods. Additionally, the requirements as noted here are subject to change as the ASME Code is revised. The appropriate edition of the ASME Boiler & Pressure Vessel Code should be consulted.
All three analysis methods are based on using numerical analysis, such as finite element analysis, to evaluate the pressure vessel.
Elastic Stress Analysis Method
With the elastic stress analysis method, stresses are computed using an elastic analysis, classified into categories, and limited to allowable values.
Specifically, linearized membrane and linearized membrane plus bending equivalent (von Mises) stresses are compared to allowable stress limits. The magnitudes of the allowable stress limits vary depending on the stress category. The different stress categories for use with the elastic stress analysis method for protection against plastic collapse are: general primary membrane equivalent stress, local primary membrane equivalent stress, and local primary membrane plus bending equivalent stress.
Limit-Load Analysis Method
The purpose of a limit-load analysis is to determine a lower bound to the limit load of a component. The allowable load on the component is established by applying design factors to the limit load such that the onset of gross plastic deformations (plastic collapse) will not occur.
A limit-load analysis performed in accordance with the ASME Code consists of using scaled loads (e.g. pressure is scaled by a factor of 1.5) and modeling elastic-perfectly plastic material behavior. If the analysis converges for the scaled loads, the design is considered acceptable.
Elastic-Plastic Stress Analysis Method
With an elastic-plastic stress analysis, a collapse load is derived considering both the applied loading and deformation characteristics of the component. The allowable load on the component is established by applying design factors to the plastic collapse load.
An elastic-plastic stress analysis performed in accordance with the ASME Code consists of using scaled loads and modeling elastic-plastic material behavior. Non-linear geometry is considered in the analysis. If the analysis converges for the scaled loads, the design is considered acceptable.
The Popularity of the Elastic Stress Analysis Method
The elastic stress analysis method is the most commonly used method of the three options. This is likely the result of a number of factors including: the method is specified by the customer (owner), the finite element analysis software being used can only model elastic material behavior, or unfamiliarity with the other methods.
While the other methods have been an option within the ASME Code for decades, the computing power required to perform limit-load and elastic-plastic analyses have often made these options less attractive in the past. Additionally, there is often a desire to use an acceptance criterion of “the maximum stress is X, which is less than the allowable limit of Y”, as opposed to “the analysis converged”.
Differences Between the Non-Linear Analysis Methods
At first glance, the limit-load analysis method and the elastic-plastic analysis method look very similar. They are both technically elastic-plastic analyses. The primary difference is the limit-load analysis method uses simplified, yet conservative, material properties for post-elastic (i.e. plastic) behavior while the elastic-plastic stress analysis method allows a more accurate material model for post-elastic behavior. Other differences include scaling factors applied to the design loads, and whether or not small displacement theory is used in calculating displacements and strains.
Post-Yield Material Behavior
The simplified material model for the limit-load analysis method is defined as elastic-perfectly plastic with the yield point set equal to 1.5S, where S is the base allowable stress of the material. The material model for the elastic-plastic stress analysis method does not have to be defined as elastic-perfectly plastic but can include hardening or softening.
With a perfectly plastic material model there is no work hardening. Therefore, a material model that includes hardening can withstand more load prior to collapse than one that is elastic-perfectly plastic; assuming the yield point is the same.
When Yielding Occurs
The yield point used for the limit-load analysis method is set equal to 1.5 times the base allowable stress of the material. The base allowable stress, S, is a fraction of either the yield strength or ultimate tensile strength of the material. The less ductile the material, the more likely the base allowable stress will be based on the ultimate tensile strength of the material.
If the base allowable stress of the pressure vessel material is based on the yield strength, the yield point for the stress-strain curve for the limit-load analysis method and the elastic-plastic stress analysis method will be the same.
However, if the base allowable stress is based on the ultimate tensile strength, the yield point for the stress-strain curve for the limit-load analysis will be lower than the elastic-plastic stress analysis method which is set equal to the yield strength of the material. In this situation, plasticity, and the onset of plastic collapse will occur sooner.
Calculation of Displacements and Strains
Another difference between the limit-load analysis method and the elastic-plastic stress analysis method is the basis for calculating displacements and strains. Small displacement theory is required for the limit-load analysis method. The elastic-plastic stress analysis method requires that non-linear geometry (large deflections) be considered.
Small displacement theory is appropriate when the displacements are less than half the thickness of the material. Displacements larger than this can affect the stiffness of the component and thus the accuracy of the results.
In a finite element analysis (FEA) that uses large deflection, the stiffness matrix is updated based on the displaced shape of the component, resulting in more accurate results. A large deflection analysis using FEA is an iterative analysis, where the effect of this change in geometry on the stiffness and resulting strains are accounted for.
Therefore, an analysis using the elastic-plastic stress analysis method has the potential to withstand more load prior to plastic collapse than the limit-load analysis method.
Scaling Factors on Design Loads
Considering how the stress-strain curve is defined and how displacements are calculated, the elastic-plastic stress analysis method would seem to be the more attractive method in that it can withstand higher loads prior to plastic collapse. This would be the case if the scaling factors on the design loads were the same for the limit-load analysis method and elastic-plastic stress analysis method. They are not.
The magnitudes of the scaling factors on the design loads for the elastic-plastic stress analysis method are approximately 60% larger than those for the limit-load analysis method.
Therefore, assuming the yield point used for the stress-strain curve is the same for the limit-load analysis method and the elastic-plastic stress analysis method, and assuming the effects of large deflection are negligible, the advantage of one method over the other comes down to whether or not the lower plastic strains with the elastic-plastic analysis are enough to offset its use of higher design load scaling factors.
Limit Load, Elastic-Plastic, or Both?
Given all of the above, it may not be that straight-forward in deciding which method (limit-load or elastic-plastic) to use. However, fretting over making this decision may not be a problem. With the exception of defining the stress-strain curve for the material model and specifying small versus large displacement theory to be used, the same finite element model can be used for all three analyses. Modifying the analysis input parameters to run one analysis method or another is usually not overly complicated or time consuming.
Computer Run Time vs Time Evaluating Results
Lastly, in comparing the three analysis options, there is a difference in computer run times and evaluation of the results.
Limit-load analyses and elastic-plastic analyses are highly non-linear resulting in longer computer run times compared to elastic analyses. Depending on the size of the model and available computer resources, the elastic stress analysis method may be more desirable.
Conversely, evaluating the results of an analysis performed in accordance with the elastic stress analysis method requires defining locations to perform linearized stress calculations and then evaluating the results. This can be time consuming depending on the number of locations requiring evaluation.
Conclusion
In conclusion, the decision on which analysis method to use to ensure protection against plastic collapse is not always straight-forward. Ultimately, it may be a matter of personal preference or up to the customer. Lastly, it should not be forgotten that any of the three methods are considered acceptable by the ASME Code. A pressure vessel design that does not meet the acceptance criteria using one method may be shown to be acceptable using one of the other methods.