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Stress Converges Slower Than Displacement

Stress concentration around a hole is a problem whose exact solution is known. A coarse mesh reads the peak stress nearly half too low — that is, on the unsafe side. And with the same number of elements, changing only where they sit cuts the error from 48% to 1%.

Confidence: A Engine: engine/fem2d.js Tests: tests/fem2d.test.js — 50/50 Prerequisite: FEM-02 Reading: ~16 min Last verified: 2026-09-10 Türkçe: bu yazının Türkçesi
Correction — 2026-09-10 The error percentages in this article are computed against the Kirsch value of 3.000. Kirsch assumes an infinite plate; the model here is finite (a/W = 0.05). For this geometry the correct concentration factor is 3.021 — shown two independent ways in FEM-06. Accordingly the true error on the 2048-element mesh is not −1.9% but −2.55%. The direction matters: Q4's error was made to look smaller than it is here, so this article's thesis — stress converges slowly, a coarse mesh reads the peak low — is strengthened rather than weakened.

1. Why this problem?

A plate with a hole is the classic teaching problem of finite elements, and for a good reason: its exact solution is known. How wrong the numerical answer is can therefore be measured rather than guessed.

But the real reason for choosing it is different. In the two previous articles we always looked at displacement. Engineering decisions, however, mostly rest on stress — and the two do not converge at the same rate.

2. The Kirsch solution

For a circular hole in an infinite plate under uniaxial tension, the tangential stress at the hole boundary is:

σθ = σ∞ · (1 − 2 cos 2θ) θ is measured from the load direction. θ = 90° (perpendicular to the load) → 3σ∞, the famous stress concentration factor. θ = 0° (along the load) → −σ∞, that is, compression.

The second value is less well known and therefore valuable: it provides an independent check. A model hitting the peak stress could be a coincidence; hitting the compression at the side at the same time could not.

3. Model and symmetry

The problem is symmetric about two axes, so a quarter model suffices. The symmetry boundary conditions are simple: on each symmetry axis the displacement perpendicular to the axis is zero.

Symmetry is not free Using symmetry divides the element count by four, but it is valid only if the geometry, the loading and the behaviour you are looking for are all symmetric. Symmetry-breaking behaviour such as buckling or crack propagation does not appear in a quarter model — the model excludes it by definition.

The hole radius is a = 10 and the quarter model half-width W = 200, so a/W = 0.05 and the finite-width effect is small; we are close to Kirsch's infinite-plate assumption.

4. Convergence

MeshElementsDOFConcentration factorError
computing…
 Grading factor 2 (mesh refined toward the hole boundary). The exact value is 3.000.
Findingcomputing…

5. Why is stress slower?

The finite element method takes displacement as the unknown. Stress is computed from the derivative of displacement:

u → ε = ∂u/∂x → σ = D·ε Differentiation lowers the order of the approximation by one. If the displacement field is represented by a polynomial of degree p, the stress is of degree p−1. In a Q4 element the displacement is bilinear; the stress stays nearly constant within the element.

Compare with the cantilever in FEM-02: at 256 elements the displacement error was 4%. Here at 512 elements the stress error is still —. A mesh of the same order gives a markedly worse answer in stress.

The practical translation: a model that appears converged in displacement may still be unconverged in stress. A mesh adequate for a deflection check is not adequate for a stress check.

6. From which side it converges — and why that matters

In FEM-02 we saw that displacement always converges from below, i.e. the model is stiffer than the real structure. Here the peak stress also converges from below — but the consequence is far more serious.

The unsafe side Displacement reading low means being optimistic in a deflection check. Peak stress reading low means believing the material has not yielded. An engineer working with a coarse mesh may see half the true stress at the hole boundary and consider the section adequate.

An important caveat: the "always from below" rule that holds for displacement is not a general theorem for stress. It behaves that way in this problem because the peak stress sits at the top of a sharp gradient and a coarse mesh smooths it. In other problems — particularly near a singularity — stress may also approach from above, or grow without bound.

7. A second, independent check: compression

At the hole boundary in the load direction, Kirsch predicts −1: in a plate under tension, compression develops beside the hole.

Meshσ(θ=0) / σ∞Error
computing…
 The exact value is −1.000. This is a second verification, independent of the peak stress.

That this value also converges correctly shows the model is not merely hitting one number but representing the whole stress field correctly.

8. Where the elements sit matters more than how many

Now the most practical finding of the article. In the table below the mesh does not change at all — it is always 32×16, that is 512 elements. The only thing that changes is how much the elements are concentrated toward the hole boundary.

GradingElementsConcentration factorError
computing…
 Same number of elements, same solver, same problem. The only difference is where the elements are concentrated.
Resultcomputing…

This is the sharpest form of a theme repeated across all three articles in the series:

In all three, the "refine the mesh" reflex was the most expensive and least effective remedy. Refining does help — but only after you have asked the right question: where is the error coming from?

9. The engine's own limit

This engine uses a dense linear solver. Memory grows with the square of the degrees of freedom, time with the cube. That is adequate for teaching models; beyond that the engine refuses explicitly rather than crashing, and says why.

In production software Commercial programs use sparse solvers: most of the stiffness matrix is zero and only the non-zero terms are stored. That is why models with millions of degrees of freedom can be solved. The limit in this article is a limit of this teaching engine, not of the method, and the result says so explicitly.

10. What to do in practice

  1. Run a separate convergence study for stress. Converged deflection does not mean converged stress.
  2. Put the elements where the gradient is. Holes, notches, corners, load application points. A uniformly divided mesh spends most of its elements where nothing is happening.
  3. Assume the peak stress is low. If you have not run a convergence study, the value you found is below the truth.
  4. Find a closed-form check where you can. Classical solutions such as Kirsch should be the first test of a new model or new software.
  5. Check at more than one point. Hitting a single number may be coincidence; hitting two different points of the stress field cannot be.
  6. Use symmetry deliberately. If the behaviour you are looking for breaks symmetry, a quarter model will not show it.

11. Test yourself

  1. Why is stress one order less accurate than displacement?
  2. Why is a low peak stress on a coarse mesh the unsafe side?
  3. Why is the stress at θ = 0 in the Kirsch solution compressive? How does that arise in a plate under tension?
  4. You built two models with 512 elements each; one is 48% in error, the other 1%. What could the only difference be?
  5. We can say "always converges from below" for displacement. Why can we not say the same for stress?
  6. A model appears converged in deflection. What would you do before trusting its stress results?

References

  1. The Kirsch solution — the stress field around a circular hole in an infinite plate; a classical elasticity result, independent of any national code.
  2. Plane stress isoparametric formulation and convergence order — standard finite element results.
  3. For course modules and workshop problems: Kwon, Y.W. & Bang, H., The Finite Element Method using MATLAB. The exposition, derivations and all figures here are our own; the source is cited for topic coverage.

All figures in this article are produced by engine/fem2d.js and separately pinned in tests/fem2d.test.js (50/50). Verification is carried out at two independent points: the concentration factor at the top of the hole (3.000) and the compression at the side (−1.000). The quarter-model mesh gives a positive Jacobian at every subdivision used, and this is separately tested.

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archi-civil.com — Stress Converges Slower Than Displacement · FEM-03 · Confidence A · Printed:
Converged deflection does not mean converged stress; a coarse mesh reads the peak stress low.