Component Calculations

Updated: 2016-12-20th MCT

A variety of components are analyzed using code rules. The results are discussed comparing the difference between component types.

Comparison Between Head Types: Hemi, SE, F&D and Flat

Quick Links (to topics in this article)

The Four Heads

Four commonly used head types on vessels are Hemispherical (Hemi), Semi Elliptical (SE), Flanged and Dished (F&D) and Flat. For this blog post, a 48” OD vessel with a 0.5000” wall (47” ID) is designed in SA-516 70 material, code rated for 20,000 psi at 100°F. The vessel is fully radiographed on long and circ seams and no corrosion is assumed. The ASME VIII-1 calculated design pressure is 420 psi (see the calculation set linked below for the full calculations).

Each of the four heads is designed to match the 47” ID of the cylindrical shell, but the head thickness is varied as required to meet the 420 psi design pressure of the cylinder.  The results:

Head Thickness [in] Outside Height [in] Volume [US gal] Weight [lbs]
Cylinder, 24” long 0.5 24 180.25 506.7
Hemi 0.2474 23.75 117.7 245.5
SE * 0.4947 13.74 70.1 397.3
F&D * 0.8901 10.29 47.7 602.9
Flat 3.9120 3.91 0 1920.8
* Including the 1 ½” straight flange
Four Heads (left to right) Hemispherical, Semi Elliptical, Flanged and Dished and Flat

Four Heads (left to right) Hemispherical, Semi Elliptical, Flanged and Dished and Flat

Hemispherical Head (Hemi)

The hemispherical head has a simple radial geometry: the depth of the head is half the diameter.  With a 47” ID, the required wall thickness is 0.2474”, about half the thickness of the shell (if the head was modelled at 47.25” ID, providing the same average diameter as the shell, the thickness would be closer to half the cylinder thickness).  Because the head is thinner than the shell, a standard code 3:1 taper is used on the transition, a part of the stronger head, the shell is not tapered down on the straight section because it needs the full thickness.

Usually a hemi head cannot be formed from a flat sheet, instead it is made from welded pieces, making this, the thinnest head, sometimes the most expensive.  It is commonly used in large diameter or high pressure applications where material savings are important.  Two spherical heads back to back make a storage sphere, the most efficient shape for pressurized storage.

Semi Elliptical Head (SE)

The Semi Elliptical head has an elliptical form – the most common ratio is 2:1 – or the width of the ellipse is twice the depth.  Other ratios are possible but not commonly used.  In practice the fabricator will often make the SE head from 3 radii that approximate an ellipse – large in the crown, smallest at the outside diameter, with an intermediate radius in the middle.  Code rules dictate how close the approximation has to be to a true ellipse.  Code rules also exist allowing a two radius – crown and knuckle – which would normally be considered as a F&D head, to be considered a SE head if special values are used (Ug-32(c): An acceptable approximation of a 2:1 ellipsoidal head is one with a knuckle radius of 0.17D and a spherical radius of 0.90D.).

This 2:1 SE head is made from half of the ellipse, so the head depth is a quarter the diameter – half the hemi head, but more than the F&D and Flat head.  SE heads can be made from a flat plate, resulting in what is often the most economical head for low pressure vessels.

The SE is not as efficient at handling stresses as the hemi, so the design rules require more thickness.  The ASME code design formulas for a 2:1 SE are very close to that of the cylinder – in this case resulting in a required thickness of 0.4947” for the SE vs 0.500”.

Flanged and Dished (F&D)

Flanged and Dished heads are commonly used where pressure is moderate and the overall height is important.  Here the 48” inside radius (equal to the outside diameter of the cylinder) along with a tight 2.973” knuckle results in a head that is lower than the semi-elliptical.  The tight knuckle radius results in high forming stresses – in this case post forming heat treatment (stress relief) is required.

The flanged and dished head requires more thickness than the matching cylinder, here 0.8901”.  Again a code standard 3:1 transition on the straight flange (Which only needs to be 0.5000”) handles the difference in thickness.  Unless the height is important, a vessel with a pressure as high as this 48” design would typically use a SE instead.

Flat Head

The hemi head is the most efficient, containing the pressure in pure tension.  The other designs substitute various amounts of bending stresses at lower efficiency to lower the head height and pay for it in increased weight.  This flat head, working purely in bending, pays for it with a massive 3.9120” thickness.  Flat heads are usually reserved for processes that require flat inside surfaces.

Many solutions have been developed to provide flat heads on the inside of the vessels with more efficient methods of handling the pressure stresses:

  • Thin flat plate with tie rods or rings connected to the SE or F&D head it is mounted in. The head supports the load, and the plate provides a flat inside surface.
  • Pouring a flat concrete floor in SE or F&D heads.
  • Thin flat plate supported by Exterior beams across the width.
  • Thin flat plate with stay rods (or tubes) through the length of the vessel to the opposite flat head.
  • Thin flat plate with diagonal stay rods tied to the shell – often seen in boilers.


Cylinder and Hemi Head Tresca Stresses

The ASME VIII-1 code equations used for Cylinders and Hemi heads are easily derived.  The ½” thick cylinder ends up with a stress equal to the code design stress limit of 20,000 psi – actual measured stresses = 20,484 psi Tresca or Stress Intensity P1-P3.  Also the 0.2474” thick hemispherical head ends up with a stress of 20,364 psi.  Both stresses are very close to the target.

Stress Intensity (P1-P3) in a Cylinder and Hemi Head equaling the code design target stresses.

Stress Intensity (P1-P3) in a Cylinder and Hemi Head equaling the code design target stresses.

The stress is higher in the discontinuity zone of the head to shell junction (23,060 psi).  The VIII-2 code rules allow for these increases over small distances and provides limits.  The VIII-1 rules, beyond the requirement for a 3:1 taper, ignore these stresses which are known to be acceptable.

Cylinder and Hemi Head von Mises Stresses

The previous section shows a very close match between the code rules and the measured FEA stresses for the cylinder and the Hemi head.  However, VIII-2 changed from using Tresca (P1-P3 stress) to von Mises methods.

Von Mises stress in the cylinder and hemi head – the cylinder stress is now 12% below the VIII-1 code

Von Mises stress in the cylinder and hemi head – the cylinder stress is now 12% below the VIII-1 code

The von Mises stress results range from equivalent to up to 15% lower than Tresca (P1-P3) results.  In this example the cylinder stresses dropped to 17,740 psi, 12% below Tresca, but the hemi head stress remains at 20,322 psi.  FEA results are required to be done to VIII-2 methods, including the use of von-Mises stress reporting, however VIII-1 stress equations that are derivable are done to Tresca methods.  Most Canadian reviewers require von-Mises stresses to be used (see ABSA guidelines), some from Saskatchewan demand Tresca, others von Mises.  ASME has been asked to interpret which stress method should be used for VIII-1 vessels and has refused to answer.  This leaves the designer stuck in the middle.  The consensus answer is to use von Mises unless asked to do otherwise.

VIII-2 has rules for the design of cylinders which match the Tresca stress methods, however, VIII-2 also allows FEA results to replace any design rule. A design-by-analysis in accordance with Part 5 may be used to establish the design thickness and/or configuration (i.e. nozzle reinforcement configuration) in lieu of the design-by-rules in Part 4 for any geometry or loading conditions (see Design Thickness. The design thickness of the vessel part shall be determined using the design-by-rule methods of Part 4 with the load and load case combinations specified in Alternatively, the design thickness may be established using the design-by-analysis procedures in Part 5, even if this thickness is less than that established using Part 4 design-by-rule methods. In either case, the design thickness shall not be less than the minimum thickness specified in 4.1.2 plus any corrosion allowance required by 4.1.4.

The designer will get a thinner shell when designing to VIII-2 part 5 than VIII-2 part 4.  As FEA methods from VIII-2 part 5 gradually replace code rules as found in VIII-1 and VIII-2 part 4, reduced cylindrical thicknesses can be expected.  The Hemi heads will not change.

The remainder of this article uses von Mises stress.

Stresses in SE & F&D Heads

The VIII-1 formulation for 2:1 SE heads results in required thicknesses equal to that of the shell.  However, the code equation is not a predictor of actual stress.  It is just a design rule that produces results that are known to be acceptable.  The actual stress in the SE head is higher in the knuckle region and equal to the design stress in the crown.  VIII-1 nozzle reinforcement rules account for this requiring more reinforcement in the knuckle.

The F&D head, even with its thicker construction, has much higher stress in the knuckle region.  It is common in thinner F&D heads to exceed the VIII-2 allowable stresses in the knuckle.  Programs like Nozzle Pro often cannot calculate nozzles in F&D heads, because the heads fail VIII-2 rules, even without the added stress of an included nozzle.  F&D heads are known to be safe, but if the heads were invented today, the required thickness for some would be higher.  Designers are particularly cautioned about putting large nozzles in the knuckle region

SE (left) and F&D head – the knuckle stresses are higher, especially in the F&D head

SE (left) and F&D head – the knuckle stresses are higher, especially in the F&D head

As FEA methods are more commonly used, it can be expected that some F&D head thicknesses (for large diameter thinner heads) will be higher.  SE head designs are not expected to change.

Stresses in Flat Heads

VIII-1 formulas for flat heads result in stresses much lower than code rules allow.  The flat head is in bending, which has allowable stresses of 1.5x membrane, or 30,000 psi in this case.  The actual center stress is half of this.  The code rules vary the allowed center stress based on the attachment method, the rules are really controlling the stress in the head to shell discontinuity zone, not the center of the head.

Very low code allowable stresses in the flat head, higher in the shell junction

Very low code allowable stresses in the flat head, higher in the shell junction

In this example, both the junction and head stresses are lower than otherwise allowed by the code.  The stress in the head is lower at the edges than the center, leading to the design of heads that taper at the edges.  These heads cannot be designed to standard VIII-1 rules, sometimes leading Canadian reviewers to insist on the use of fully flat heads, at which point code rules can be applied, and further once applied considerably heavier heads are required.  Insistence on the use of VIII-1 rules where not expected is unpleasant, but sometimes a fact of life in Canada.  As FEA methods become more common, expect flat head thicknesses to reduce.

Calculation Set

Download the Compress ASME calculation set for the four heads and cylindrical shell.

Flanged and Flued Expansion Joints

File:PVE-4305, Last Updated: June 22 2010, By: LB
ASME VIII-1 mandatory Appendix 5 provides guidelines for the design of flanged and flued expansion joints, but does not provide methods of calculating the stresses, fatigue life or spring rate. ASME Appendix 26 (and EJMA) provides rules to calculate these values, but the configuration of a flanged and flued expansion joint does not match that used in appendix 26 or EJMA.

Here the rules of Appendix 26 are applied to an Appendix 5 flanged and flued expansion joint. The calculated results are compared to finite element analysis which shows that using the Appendix 26 calculation method is conservative.

Flanged and Flued Profile

Cross section of a flanged and flued expansion joint


Flanged and flued expansion joint

Flanged and flued expansion joint

The flanged and flued expansion joint has a straight crown and straight cuff neither of which are included in the ASME/EJMA stress and flexibility calculations. These straight sections affect the stresses, spring rate and cycle life. Here an expansion joint with the dimensions Db=24, t=1/4, w=3 and q=3 is examined by FEA with and without straight sections. The pressure used is 300 psi and the bellows is compressed by 1/2 inch.

Standard profile bellows

Profile of a bellows that can be analyzed by ASME/EJMA methods – no straight crown or cuffs are included. The section is swept 1° for analysis


Flanged and flued expansion joint

Stress results for the standard shape at 300 psi and 1/2″ compression. Maximum stress is 262,000 psi.

The initial stress results show a high stress of 260,000 psi which compares with an EJMA computed stress of 215,000 psi (see link below for the EJMA calculation results). The stress indicates that the bellows is operating above the yield point so the EJMA predicted spring rate will be too high. Comparing the accuracy of FEA vs EJMA methods is beyond the scope of this article.

The addition of the 1″ straight crown lowers the stress to 249,000 psi:

Addition of a straight crown

Addition of a 1″ long straight crown


Flanged and flued expansion joint

Stress results for the modified shape – same pressure and displacement. The stress has lowered to 250,000 psi.

The further addition of straight cuffs lowers the stress more:

Standard profile bellows

Straight cuffs have been added to the bellows


Flanged and flued expansion joint

Stresses have dropped to 175,000 psi

The bellows stress has now been lowered to 175,000 psi, less than the EJMA calculated 214,000 psi. The use of EJMA methods for this flanged and flued bellows is conservative and acceptable. The EJMA calculated spring rate will be too high and the theoretical cycle life too low both of which are conservative errors. See the link below for the EJMA calculations for this expansion joint.

Downloads (pdf format)