File: PVE-3179, Last Updated: Dec. 13, 2008, By: LRB


Error plots show how well the complexity of a mesh matches the complexity of the deflections in a model. Once the mesh complexity matches the model complexity, the reported error is low. As a guideline, Pressure Vessel Engineering uses 5% error as an acceptance criterion.

It is possible to get stresses below 5% in general vessel areas by applying an appropriate mesh size. This report covers two areas where the error cannot be lowered to reach this acceptance criteria regardless of the mesh size used. These areas are: 1) stresses in and around the head of a bolt and 2) stresses at surface to surface contacts.

Other areas also exist in pressure vessels where mesh refinement can not be used to reduce errors to this 5% acceptance level. These areas: weld fillets, diameter transitions, nozzles, flanges and support legs and lug attachements are beyond the scope of this article.


Solid model of test shape

Example test shape – an assembly of 3 parts: 2 plates of 2″ x 2″ x 1/2″ thick with 1/2″ radius hole in one corner. The test plates are joined with a 7/8″ root diameter bolt. The bolt is made 0.002″ shorter than the two plates to create an interference fit preload.

Model with no penetration surface defined.

A no penetration surface is defined between the two plates. The plates can separate but not pass through each other.

Example of interference fit

An interference fit is defined between the bolt head and the top plate. The bolt will be stretched to reach the top surface. The top surface will be compressed by the bolt.

Model with symmetry boundary conditions applied.

Symmetry boundary conditions are applied to sides and bottom of assembled model.

Model with mesh at 1/8 size.

The model is meshed at 1/8″ size.

Close-up of model with interference mesh.

Close up of the interference mesh between the bolt and the top plate.

FEA Model of displacement plot.

Displacement plot – the plates are in contact under the bolt head, separated elsewhere. The bolt was stretched > 0.01″ to create a preload. The plate also compressed under the bolt head. This stretch is shown magnified x125 here so the bolt appears to be out of contact with the top plate – it is in contact.

Close-up of model showing contact area.

A close-up of the contact area between the two plates.

Intensity stress plot.

Intensity stress plot (Tresca P1-P3 criteria) – the highest stress is indicated under the bolt head.

Close-up of FEA model showing stress area.

Close-up of the highest reported stress area – under the bolt head.

Overall error plot.

Overall error plot. The error plot shows areas in and around the bolt head to be higher than the 5% acceptance criteria.

Error plot scaled to 100%.

The error plot scale re-scaled to 100% maximum. The maximum error is located under the bolt head at the edge of the bolt to top plate interference contact. The sharp edge of the contact area can not be eliminated regardless of the mesh size used. This area will always have a high indicated error.

ASME VIII-2 (2287 Ed.) sets the stress limits for bolts at locations away from the stress concentrations.

VIII-2 5.7.2(a): The maximum value of service stress, averaged across the bolt cross section and neglecting stress concentrations, shall not exceed two times the allowable stress values in paragraph 3.A.2.2. of annex 3.A

VIII-2 5.7.2(b): The maximum value of service stress, except as restricted by paragraph [fatigue assessment of bolts] at the periphery of the bolt cross section resulting from direct tension plus bending and neglecting stress concentrations shall not exceed three times the allowable stress values in paragraph 3.A.2 of Annex 3.A

The bolts are studied at some location other than under the head. Large stress concentrations are also created at the location where the bolt threads into its parent material (not shown in this model). This area will also show a high indicated error.

Model with area of high reported stress.

Another area of high reported error: the contact between the top and bottom plates.

Close-up of model with contact pressure.

Close-up of the previous shot – contact pressure at the surface to surface contact between the two steel plates. These contact areas show as high errors regardless of the mesh size used.