Non-Linear (Elastic-Plastic) Analysis
File: File:PVE-4433
Last Updated: Aug. 12, 2010
By: DRV
Non-linear analysis is an elastic-plastic analysis method to determine if plastic collapse will occur. It is a more accurate assessment than a linear elastic analysis as it accounts for yielding and the redistribution of stress that occurs in a component as a result of plastic deformation. This example provides a comparison of a linear and non-linear analysis. Advantages and disadvantages of each are discussed and recommendations for their use provided. A complete engineering report is available below representing a typical non-linear analysis and report provided by Pressure Vessel Engineering.
 Elastic-Plastic Material Curve |
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This graph compares the modulus of elasticity used for a linear and non-linear analysis of the same material. A linear analysis assumes that the material is perfectly elastic. As a result the analysis does not account for the effects of yielding and reports much higher stresses past the yield limit opposed to a non-linear study. A non-linear (elastic-plastic) analysis also assumes the material to be perfectly elastic up to the yield point, providing results identical to a linear solver. However past the yield point the redistribution of stress as a result of plastic deformation as well as the effects of strain hardening are accounted for. This results in higher deformation and lower reported stresses than a linear analysis.
 Linear Material Model - A maximum displacement of 0.018" is observed (displacements are magnified) |
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 Elastic Plastic Model - A much higher, more realistic displacement of 0.044" is observed (displacements are magnified) |
 A maximum stress of 41,698 psi is observed |
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 A lower, more realistic maximum stress of 36,011 psi is observed. |
So why not always use a non-linear solver if it is more accurate? A linear analysis reports the instantaneous effects of the applied boundary conditions and loadings on the component. As a result only a single computation is required to obtain the results. To analyze the effects of plastic deformation a non-linear solver must run numerous iterations. Past the yield point the component starts to plastically deform and permanently change shape. The non-linear solver accounts for this and runs as many iterations as required to accurately update the displacement of the component and the effects this has on the applied loads. For this reason a non-linear study requires tens, hundreds, even thousands of times the computations a linear study would require.
Results
As a linear analysis assumes the material to be perfectly elastic it will "overestimate" stresses in excess of the yield point. To overcome this issue allowable stresses are established for different regions of the component. For example ASME VIII-2 will only allow 1X the listed material stress for general areas where plastic collapse would be expected (center of a flat head), but up to 3X (well above the yield and approaching the tensile strength) for local regions which are expected to plastically deform and "self-limit" themselves.
Since the flange experiences yielding in the hub area, the model remains in a slightly deformed shape even after the loads have been removed. |
Recommendations
For most analysis a linear solver is adequate. It reports identical displacements and stresses to that of a non-linear solver up to the yield limit and is much more efficient in doing so. If stresses are in excess of the yield point allowables can be established to provide acceptance criteria.
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