FEA Stress Analysis

Pressure Vessel Engineering is an industry leader in applying Finite Element Analysis (FEA) techniques to the design of pressurized equipment. We have run over 1,500 FEA based reports validating product safety or illustrating shortcomings. Our reports are in compliance with ASME VIII-2 and can be made to comply with the multiple different Canadian provincial registration guidelines. Our FEA capabilities include Linear Elastic Analysis, Non-Linear Elastic Plastic Analysis, Permissible Cycle Life Analysis, Buckling Failure Safety Analysis, Steady State and Transient Thermal Analysis, Flow Analysis and Frequency & Vibration Analysis.

FEA Articles


Finite Element Analysis in Action

What is Finite Element Analysis (FEA)? When can it be used? Many of our customers are only vaguely aware of how useful this emerging tool can be in solving their design problems. If you have never heard of it before you are not alone. This short pictorial shows FEA in use on what at first glance looks like a simple design problem. Mechanical engineers have used Finite Element methods as long as computers have existed. In the last 15 years personal computers have become powerful enough to solve the very complex equations used in the analysis. Along the way the software has become easier to use making FEA a more widely used tool. This sample uses our favorite FEA software, SolidWorks Simulation (SWS) and SolidWorks Flow Simulation, to design a cast pipe bend.

A Design Problem

This is what the customer wants: a tight radius thick wall cast pipe bend. The customer has requested the bend to have the same pressure rating as the maximum rating of the pipe it is welded to. Here it is shown with the same wall thickness as the pipe, but the bend wall thickness might need to be adjusted depending on its stresses. How can this casting be made to work? (Pipe OD = 1.5″, matching pipe wall thickness = 0.375″, leg center to center distance = 2″, pressure rating of the attached 3/8″ thick pipe = 11,090 psi.  The allowed stress rating of the cast material is 20,000 psi.)

A pipe bend with tight radius and a thick wall.

A pipe bend with tight radius and a thick wall.

Code Rules

Pipe bends are widely used in piping design and mandatory code rules provide the required wall thickness and pressure rating, but there are limitations.

pipe bend code rules

ASME code rules covering the required wall thickness of a pipe bend. The location of the intrados and extrados are shown.

When this bend is calculated to the above code rules from ASME B31.3, two errors pop up: 1) the thickness to diameter range is too large (another way of saying the leg spacing is too tight), and 2) the Pressure is too high to use a thin wall equation. Out of interest, the rules do calculate that the intrados (inner radius) should be 2.2x thicker than the extrados (outer radius). (The calculated minimum intrados = 0.768″, calculated extrados = 0.335″ but the rules cannot be used.)

The FEA Process

With no mandatory code rules available, FEA or burst testing can be used. Although we might choose to use burst testing to validate the FEA results, burst testing on its own cannot provide the level of knowledge gained from FEA.


Simplification of the problem through two fold symmetry.

The first step of the FEA analysis is to build a solid model. The bend is symmetrical top to bottom and left to right. The loads are also the same top to bottom and side to side so the stress patterns will repeat.  This allows a valid simplification of only studying 1/4 of the bend.  To start, the bend is made the same thickness as the pipe it is attached to. To improve accuracy of the results, the length of the bend is increased to include some of the attached piping. In this case SolidWorks (SW) is used to model the bend.

Mesh with pyramids

Mesh of 20,400 elements. Typical pyramid shaped elements are shown.

The solid model is imported into SolidWorks Simulation (SWS) where it is meshed into small triangular base pyramids known as elements. In this case 20,500 elements are used to fill the volume of the bend.

boundary conditions

Boundary conditions.

Symmetry boundary conditions have been applied to the three cut faces.  The blue body shows the equivalent shape that the program thinks it is calculating if the bend is mirrored on each boundary surface. The boundary conditions make it easier to see inside the model during analysis and speed up the program execution.

applied pressure design

Applied design pressure.

The design pressure of 11,090 psi is applied to the inner surface of the model.

fea equations

FEA equations for each node of the element.

SWS is not a pressure vessel calculator or a pipe bend calculator. It is a general physics simulator. It does not know anything about pipe bends, but it does know how to analyze the deflection and stress of any pyramid shaped element it is given. It applies the above equations to each node of each element in the model – a total of 82,000 times for this small example. For each node SWS calculates the displacement and then the stress. The trick is that it calculates all the elements in the mesh at the same time and gives the overall results for the whole study, whether it is a pipe bend or any other object that can be modeled. This sample is using 20,500 elements, but many models we run use several million. The great thing about FEA is that the operator has no need to get involved with the really hard math the computer is doing.

displaced plot

Displacement of the bend under pressure magnified 2000x.

Displacement is calculated first. Here the displaced shape has been magnified 2000x. It can be seen from the top view that under pressure the legs separate. The diameter also grows, but not from the centerline. The displaced shape is complex enough that the analyst cannot easily predict it. From this it is expected that the stress pattern will also be complex.

stress plot

Stress plot with area of excess stress shown. The design fails.

Next the stress. Pressure vessel FE analysts are skeptical about many ASME code rules, but here is a case where the rules relate to the FEA results. The code rules predicted more material would be required at intrados as shown by the arrows where the stress has failed.

Design Alternatives

The designer needs to come up with some approaches to solving the problem. Often it takes multiple attempts to gradually home in on a few good alternatives. Fortunately we can do this without too much difficulty. If the job is not a rush the analyst can break the job over several days and gradually figure out some elegant insights by experimentation. If the job is a rush, perhaps the most elegant approach might not be taken, but still a solution can be found.

base design

The base design: Fail.

Here is the base design again. The model shape, displacement and stress plots are all on one image. Below some alternatives separate ideas that work from ones that fail. The same displacement multiplier and stress scale is used for all.

outside reinforcing

Outside reinforcing – horrible results

leg bracing

Bracing the legs togeth – another horrible idea.

The first two ideas attempt to lower the stress in the intrados by reducing the separation of the legs. The outside reinforcing makes the situation worse, the leg bracing does not help. There is still too much stress at the intrados and the extrados is overbuilt. FEA does not care how smart you think your design ideas are. Either they pass or fail. Here the ideas of reducing the stress through controlling the displacement of the legs has failed. The analyst has to try again, but for this model, each run does not take much time. Soon better ideas start surfacing.

Intrados Reinforcing

Intrados reinforcing.

intrados reinforcing

Two attempts to reinforce the Intrados. Almost a pass…

A different approach is to add more material near where the stress is the highest. The approach fails in both cases, but “intrados Reinforcing 2” is close to a solution.  The analyst is starting to get better ideas.  Perhaps there is not enough reinforcing material?

heavier wall

How about a lot more material? At last a pass!

Never underestimate the ability of extra material to solve a pressure vessel design problem. Quick and crude – two things are apparent: 1) the stress in the intrados has been reduced to an acceptable level and 2) there is a lot of material in the extrados and other areas with low stress levels.  The analyst suspects the excess material in low stress areas is not helpful and considers other ideas.  The design is workable, but can we do better?

heavier wall

Less material on the extrados? Also pass.

The last attempt had lots of material in low stress areas. Here some low stressed material is removed from the extrados. This design does not take up as much room or weigh as the previous attempt. It is acceptable as far as stresses are concerned.  Will the customer like it?  Is their enough time for more alternatives?

relocated flow passage

Relocated the flow passage. A pass.

If the analyst has enough time to experiment and think, more solutions start to appear. Here instead of adding more material to areas on the exterior of the bend, the inside flow passage is relocated towards the extrados. This elegant design passes with the same amount of material as the original failing design, but the material is where it is required.  The analyst likes this approach but does not know if the changes in the interior flow passage will affect its use.  CFD can tell which is better.

Computational Fluid Dynamics

The flow characteristics of the two elbows can be compared using Computational Fluid Dynamics (CFD). At PVEng we use SolidWorks Flow Simulation. The design criteria is simple – the better elbow will have a lower pressure drop. These two elbows are compared in depth in our validation set Product Design by Comparative Pressure Drop.  What follows is a summary:

The two elbows ‘Heavier Wall 2’ and ‘Relocated Flow Passage’ are put into one CFD study and run side by side.  Initially a very coarse mesh is used on both elbows.  The pressure drop across both elbows is recorded as the mesh was refined.  On average the ‘Relocated Flow Passage’ design has a pressure drop 9% higher than in ‘Heavier Wall 2’.  The conclusion is that the ‘Heavier Wall 2’ has better flow characteristics.  Pictures follow.

The initial mesh (Mesh 1) for the two passing elbow designs run in the same study: ‘Heavier Wall 2’ on left and ‘Relocated Flow Passage’ on the right. This initial mesh is too coarse to produce usable results, but it is a good starting point for the CFD program which automatically refines it where required.  The inset shows a detail of mesh 6, the final mesh size used. The CFD program has refined the mesh where required. Note that these flow cells are cubes, not the triangular elements used previously for the structural analysis.

Pressure distribution in the two elbows for mesh 6. The ‘Relocated Flow Passage’ has a more complex flow pattern.

The ratio of the pressure drop in the two elbows. The ratio is above 1 showing the ‘Relocated Flow Passage’ has a higher pressure drop. Mesh 1 and 2 are too coarse to use.  The average ratio of runs 3-6 is 1.09.  The Relocated Flow Passage has 9% higher pressure drop.

The results are clear: relocating the flow passage is a bad idea.  

Further Design Iterations 

Once the original model is made and setup it is fairly easy for an experienced analyst to run a series of design iterations looking for practical solutions. New ideas are created and analyzed and some work. Decisions are made on hard pass / fail criteria.  This is the power of FEA and CFD. Here in a short period of time the analyst went from adding material where it made the situation worse (Outside Reinforcing) to putting it where it was useful (Heavier Wall and Heavier Wall – 2). The idea of relocating the flow passage originally looked good, but as often happens in FEA/CFD design turned out to be a bad idea, but was discovered early in the design cycle.  The designer now knows not to relocate the flow passage, but is it possible to relocate the material on the outside without touching the flow? The designer could easily continue refining the design based on real data.

To repeat a key concept, FEA and CFD are not just pressure vessel design tools. They are general physics simulation methods used in many industries. In the FEA Samples and FEA Notes sections you can see how ASME code rules are applied to FEA methods to ensure the safety of pressurized equipment. Another introductory sample: A Step by Step Introduction to FEA is suggested next.


Like any specialized field, FEA for pressure vessels is full of TLAs and other weird words:

  • TLA = Three Letter Acronym.
  • FEA = Finite Element Analysis.
  • CFD = Computational Fluid Dynamics.
  • Cell = A 6 sided cubic shape used to make a solid mesh for CFD
  • SWS = SolidWorks Simulation – the FEA package built into SolidWorks and our favorite.
  • SW = SolidWorks – a versatile and powerful solid modeling package.
  • Element = A 4 faced pyramid used to make a solid mesh for FEA.
  • Node = a computational location in the element, sometimes the corner.  
  • Intrados = the sharp radius inside surface of a pipe bend.
  • Extrados = the large radius outside surface of a pipe bend.
  • PVE or PVEng = Pressure Vessel Engineering!