File: PVE-407, Last Updated: June 2, 2003, By: LB

### The Problem:

The process in this vessel required a reverse dished head. The reverse dished head could not be fabricated thick enough to meet the ASME VIII-1 rules. The chosen solution was to reinforce the head with ribs to prevent snap through.

Various alternate methods of analysis are shown here. Only the plate analysis was used for the actual job. However, the comparison of the various methods is educational.

The head diameter and thickness and design pressure of 75 psi is the same for all of the examples bellow. The material has a yield strength of 30,000 psi and an allowed design stress of 20,000 psi. The maximum allowed membrane (tensile) stress is 20,000 psi, 30,000 at regions of discontinuities. The maximum allowed membrane + bending stress is 30,000 psi, 60,000 psi at discontinuities.

### Analysis – 2D Axisymmetric with Linear Material Properties:

This is one of the simplest methods of analyzing this vessel. A cross section of the head without reinforcement is analyzed. Algor assumes that the 2D drawing is symmetrical about an axis (axisymmetric). The results show the stress distribution in the head if there is no material yielding (linear material properties).

Cross section of the reverse dished head (from center to left side). Stresses are shown for an interior pressure in this and the following shots.

The peak stress is 54,000 psi in the knuckle region, well above the 30,000 psi yield point. This head fails the ASME VIII-1 code calculations for exterior pressure, but the stresses in the knuckle region are less than the discontinuity stress limit of 60,000 psi. Predicted deflection is 0.15 inches (not shown). Perhaps the head is safe? The ASME code calculations provide a safe pressure of 57 psi for a regular dished head. Also, the use of regular dished head exterior pressure calculations is not proven for a reverse dished head.

### Analysis – 2D Axisymmetric with Non-Linear Material Properties:

This analysis allows for material yielding. The same cross section is analyzed, but for this analysis, the pressure is applied in steps, and the material will be allowed to yield (Non-Linear). The results can be seen in this movie.

Up to 64 psi, the head can be seen deflecting linearly under pressure. At 69 psi snap through is beginning (and the deflection is greater than the material thickness). At this point the head has started permanent deformation – it will not return to the original shape after the pressure is removed. Pressures beyond 72 psi show rapid snap through. The final frame shows the fully snapped through shape at 72 psi. This shape is kept permanently after the pressure is removed.

Defection of the center of the head vs pressure. Snap through starts around 66 psi.

Original and final shape of head. Loaded to 75 psi and Pressure released.

### Analysis – 3D Plate Analysis:

Reinforcing ribs were put on the head to prevent snap through. 3D analysis is required to calculate the stresses. A surface model was created in SolidWorks. The material thickness is specified at time of analysis in the Algor FEA program.

Plate model – top view – created in SolidWorks.

Plate model – bottom view.

The FEA analysis of the head in Algor showed that the stresses were acceptable. The maximum allowed membrane (tensile) stress is 20,000 psi, 30,000 at regions of discontinuities. The maximum allowed membrane + bending stress is 30,000 psi, 60,000 psi at discontinuities. Peak stresses around stress concentrations can be larger.

Membrane Stress – limited to 20,000 psi except in areas of discontinuities. At areas of discontinuities, membrane stress can be 30,000 psi. This plot shows maximum membrane stresses at 42,000 psi at a concentration which is acceptable.

Total Stress (Membrane + Bending) – limited to 30,000 psi except in areas of discontinuities. At areas of discontinuities, membrane stress can be 60,000 psi. The total stresses are acceptable.

### Analysis – 3D Solid Analysis:

A solid model was created in SolidWorks including the reinforcing ribs and all weld fillets. The actual material thickness was modeled. This was not done for the original analysis, but is included here for educational purposes.

Solid model – bottom view

Solid model detail – meshed at 1/8″ mesh size

Top side stress

Bottom side stress detail

The solid model maximum calculated stresses are found in the same location as for the plate model, but are much lower. The solid model accounts better for the stresses at connections, and allows the effect of weld fillets to be included.

The maximum stress is 28,000 psi, found in small peak areas. This value could be used with a fatigue analysis if required. All of the general stresses are below the 20,000 tensile limit, so no stress linearization is required to separate membrane and membrane + bending loads.

Snap through analysis results for the solid bottom head. pressure at 1 sec is 75 psi. At 3.5x operating press the head starts to yield.

Displaced head at 5x operating pressure – displacement magnified 2x.

### The Solution:

The design with the reinforcing ribs was successfully used. A report interpreting the results according to ASME VIII-2 rules allowed the vessel to be registered. A later modification to the process allowed a less expensive double wall head to be used instead.

### Comparison of Methods Shown:

The Solid and Plate analysis methods here produced almost identical stress results except at attachments. The Solid model with the weld fillets gave more realistic and lower stress results. The solid model was also easier to make than the plate model which required each surface to be split at all intersections. If the stresses were higher in the solid model, stress linearization would have been required to separate the membrane and membrane + bending stresses. The solid model stress linearization is more difficult than reading the stresses off of the plate model.

### Credits:

This tank was built by Price Schonstrom Inc., 35 Elm Street, Walkerton, Ontario, Canada, N0G 2V0