The most important question
Finite element analysis for welds when you don’t know the right questions
Somewhere in Finland
All data appearing in this work are fictitious. Any resemblance to real data, living or dead, is purely coincidental
Act I — Running into a wall and having fun
With my ears a bit red, I sat and observed something being done in about 50 minutes that I had run my head into for around 2 days give or take. I replicated it in half the time and got similar results. Then I started reading more on a fellow called Venant, the Mercatos problem in cartography, tensor mapping, vector projection over curved surfaces, FEA in general. All the while wondering why in 7 years if education I had been studying matrix math, mechanics, material behavior and FEM tools, but I had never actually drawn useful connections between it all. At leastI’d never made conscious note of any correlations.
Some of it is simply use it or lose. How often you practice a skill in a domain as broad as mechanical engineering, some of it shortcomings in a what I would call a philosophical framework.
Act II — Not even not even tree adjacent
So — job was to replace an existing welding solution with a new one. Not an uncommon problem to be sure. For one reason or another a process cant be used in production anymore. Loss of skill, cutting costs, these things happen. I had asked one clarifying question — “we need to achieve equivalent stresses in the weld with a different process and design?” Being answered in the affirmative, I set to work.
Two. Days. For two days I iterated and obsessed over weld stresses from contracting metal. Its worse because I enjoy this kind of work. Nothing was wrong in my mind, I knew the relatively simple simulation package I had was not meant for this and at best I could do is have comparative results. Meaning estimate the stresses from the current weld design, then replicate it with the new design the exact same way. All we really know is that the current weld works, we don’t actually know the internal stresses from cooling. It just made sense. After about the 16th iteration of it not working the way I visualized I decided I’d go back to the well and ask a master i.e my boss.
What followed was me managing to confuse somebody who is generally is not confused and them simply doing the FEM design live to see what the issue was. After about 12 minutes I realized, I had been trying to do something the tools really weren’t meant for. Even more so than I realized.
I had over’thunk’it dramatically trying to make it work. Trying to simulate internal weld stresses as well as stress from the modes of the system. Which not only is extremely complex, but it is also not relevant — the weld cooling, not modes. Recall that the idea was to have a comparative study between two solutions. What was needed was simple displacement at a distance according to the Venant’s principle (something to discuss later) , to alleviate infinities and projections issues.
What I had asked of the designed FEM was the equivalent of a thermodynamic CFD using input energy and expansion coefficients of metals, known cooling and contraction rates and I would of gotten away with it to if it weren’t for that meddling Venant and the realities of what FEM actually does.
Act III — What it does
There is a fine line between being clever and getting more out of simple tools and simply barking up the wrong tree for days on end. FEM models, tools, methods exist to approximate complex matrix mathematics with real geometry — and the approximation is almost always good enough, which is precisely why it is dangerous when it is not.
FEM converts a continuous stress field into a system of linear equations solved at discrete nodes. The mesh is not a model of the geometry. It is a model of the model — a collection of simple shapes whose combined behaviour approximates the behaviour of the real object. Within each element, the stress varies according to a polynomial chosen when the element type was designed. Quadratic elements can represent parabolic stress variation. Linear elements cannot even do that. No element type can represent a stress field that goes to infinity, which is exactly what the stress field at a sharp corner does.
The accuracy of the solution depends on two conditions. First: how well the element geometry approximates the real geometry. Second: whether the physics being modelled stays within the assumptions the element formulation was built for. Both conditions fail at a weld toe. The first because the weld toe is a curved surface approximated by flat element faces — and the normal vector of a flat face diverges from the normal of the surface it is approximating, which rotates continuously along any curve. The second because the stress at a perfectly sharp corner in a linear elastic material is not large. It is infinite. The element is being asked to represent something its polynomial cannot express.
These are not software errors. They are the correct mathematical consequences of using a discrete approximation at a point where the continuous answer is singular. The software does not know this. It solves the system of equations it was given and reports the result with the same decimal precision it uses everywhere else. The colour gradient looks like a result. It is a result — of the mesh density, not of the physics.
The engineering response to both failure modes is not to avoid FEM. It is to know which kind of wrong you are looking at, and whether it matters for the question you are trying to answer. A stress that diverges as you refine the mesh is telling you something: you are standing at a singularity, and the tool cannot help you here. Move. Read the stress somewhere else. Come back to this location by extrapolation from the stable zone — which is what Saint-Venant’s principle permits, and what the IIW hot spot method formalises. The line between clever making-do and barking up the wrong tree is not about the tool. It is about whether you know where the approximation stops and you are still standing inside it.
Act IV — it is about geometry
Take a piece of paper. Hold it flat and push down in the centre. The paper bends, the stress distributes across the whole sheet, nothing concentrates dramatically. Now fold it once and crease the fold sharply. Push down on the folded edge. The stress concentrates at the crease. Fold it again along the same line. The crease gets sharper. The concentration gets worse. Now imagine folding it infinitely many times — a mathematically perfect sharp corner, with no radius, no material yielding, no deviation from the geometric ideal. The concentration becomes infinite. Linear elastic theory, which is what FEM uses as its default physical model, predicts exactly this.
The stress near a sharp notch follows:
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σ ≈ K / √r
where r is the distance from the tip and K is a stress intensity factor that depends on geometry and loading. As r approaches zero, the stress approaches infinity. This is not a numerical artefact. It is the correct solution to the equations of linear elasticity for this geometry. Williams derived it in 1952 and it has not changed since.
FEM cannot compute infinity. What it does instead is compute the stress at the nearest node to the tip. With a coarse mesh, that node is far from the tip — r is large — and the reported stress is finite and looks plausible. With a fine mesh, the node is closer — r is smaller — and 1/√r is larger. Refine again and it grows again. There is no convergence because the correct answer is infinite, and every refinement is just approaching it from further away. The result is a number that is a function of the mesh density, not the physics. Report it and you have reported the mesh, not the weld.
The second failure mode is quieter. A flat quadrilateral element has a single outward normal — a single vector perpendicular to its flat face. On a curved surface, the true outward normal rotates continuously as you move along the surface. The element’s normal and the surface’s true normal are identical only at the element centre, and diverge everywhere else. The stress on the element face — calculated by multiplying the stress tensor by the element’s outward normal — is the correct stress for a surface oriented in that direction. But the surface is not oriented in that direction. It is curved. The traction vector points in the wrong direction because the projection is being made onto a face that does not match the geometry. This produces fringe patterns at curved boundaries — oscillating stress values at adjacent nodes that should, on a smooth surface, be smoothly varying. They are not smoothly varying because each element is reporting stress on its own face rather than on the surface it approximates.
Both failure modes are the same problem wearing different clothes. A discrete set of flat shapes is being asked to represent a continuous curved reality, and at the specific locations where the continuous reality is either singular or rapidly rotating, the discrete approximation fails visibly. The singularity announces itself — the numbers grow with every refinement. The curvature problem does not — the numbers are wrong from the start and stay approximately wrong as you refine, which is worse, because the wrong answer stabilizes into something that looks convincing.
Neither failure makes FEM useless. Even my own results for the two days of effort were technically usable. Had I the insight to interpret the results.
Act V — Woulda, shoulda, coulda
Moments like these make me question if I am even in the right profession. Surely, someone with years education and some practical experience to boot should not be making these mistakes.
The issue really is the correct mental model. Granted, in time it would become automatic, a habit, but without it no amount of education will make a difference. Does not matter how many hammers you own if you cant tell a screw from a nail. There are any number of checkpoints one might consider, but for me it is understanding that matters most. Perhaps some of the onus is on the issuer of the problem to be solved, but I personally tend to take ownership of my mistakes and focus on the one thing I can control — my own mind. Everything else is up to the universe.
Act VI — The cost and benefit
The cost of all days was days, it could have been mitigated by me simply asking what the specific expected result should be is ala “What are the outcomes we are looking for”. Nothing mind-blowing. Chances are I might of had similar epiphanies along the way had I done so. It is a lesson in effective communications and assumptions. Then again, human memory being the way it is, I am not even sure you can learn engineering in any other way — not effectively or permanently in the sense of not repeating the same mistake over and over. Though humans being what they are, some mistakes will be repeated until the lesson sinks in. In some sense it is not the work being slowed down, it is the work
Epilogue
I have been prompted to ask questions since first year at university. Usually by “Any questions?” after an explanation or monologue. I always had a lot of questions. Generally as a means to the end of narrowing my own ignorance. Unknown terminology, concepts, derivations. As a working engineer however, I’ve realized the most useful questions are to clarify assumptions and expected results.
Asking the right questions to enable you to solve the right problem is one of those duh’ moments that seems obvious as obvious can be in hindsight, but how? For me it seems to be taking reality, boiling it down to first principles and then only adding the bare minimum complexity to satisfy the constraints of being a model of your problem. It also occurs that I should probably write between the difference of FEM studies and the discipline of FEA. As well as IIW hot spot stress methods.
References
**Cited
- Williams, M.L. (1952). Stress singularities resulting from various boundary conditions in angular corners of plates in extension. Journal of Applied Mechanics, 19(4), 526–528. — the paper that first derived the 1/√r relationship at a re-entrant corner. Short, readable, foundational. This is the mathematical source for everything in Act IV.
- Hobbacher, A.F. (2016). Recommendations for Fatigue Design of Welded Joints and Components. IIW-2259–15. Springer. — the source for the 0.4t and 1.0t extrapolation distances and the FAT class system.
- Niemi, E., Fricke, W. and Maddox, S.J. (2006). Fatigue Analysis of Welded Components: Designer’s Guide to the Structural Hot-Spot Stress Approach. Woodhead Publishing. — the practitioner’s guide to what you should have done instead.
**Further reading
- Zienkiewicz, O.C. and Taylor, R.L. (2000). The Finite Element Method: Volume 1 — The Basis. Butterworth-Heinemann. — chapters 1 and 2 for what FEM actually is as a mathematical object. Not to read fully — to find the one paragraph that explains why element formulations have embedded physical assumptions.
- Timoshenko, S.P. and Goodier, J.N. (1970). Theory of Elasticity. McGraw-Hill. — the stress concentration chapter. Gives you the K factor in σ ≈ K/√r in physical rather than numerical terms.
- Barré de Saint-Venant, A.J.C. (1855). Mémoire sur la torsion des prismes. Mémoires des Savants Étrangers, 14, 233–560. — cited for completeness and because the man deserves his name in your references. Not practical to read in full. The principle is better accessed through Timoshenko’s discussion of it.
- Maddox, S.J. (1991). Fatigue Strength of Welded Structures. Abington Publishing. — background on why weld fatigue requires different treatment from parent metal fatigue.
