Plane strain vs axisymmetric: the depth the stress reaches
Plane strain and axisymmetry are the two two-dimensional idealisations available in geotechnical finite element analysis, and choosing between them is not a modelling convenience — it changes how deep the load is felt. Under a uniformly loaded strip the vertical stress increment is still 10% of the applied pressure at 6.34 times the width, while under a circle of the same width it has fallen to 10% by 1.85 widths, a factor of 3.42 in depth of influence. This article computes both from the Boussinesq solution and states the consequences for mesh depth, layer selection and settlement.
Two-dimensional analysis offers exactly two idealisations of a three-dimensional world. Plane strain assumes the geometry and loading extend unchanged to infinity in the out-of-plane direction, so the out-of-plane strain is zero. Axisymmetry assumes the geometry is a body of revolution about a vertical axis, so nothing varies with the circumferential coordinate.
An embankment, a long retaining wall and a tunnel are plane strain. A circular tank, a pad footing and a single pile are axisymmetric. So far this is a modelling convenience.
It stops being a convenience when you notice that the two carry load into the ground in fundamentally different ways, and that a strip load does not decay with depth the way an engineer’s intuition — trained on footings — expects.
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The two closed formsPermalink to “The two closed forms”
Under the centre of a uniformly loaded circle of radius on an elastic half-space, the vertical stress increment is
Under the centre of a uniformly loaded strip of width , with the angle the strip subtends at the point,
Both are Boussinesq results — linear elastic, homogeneous, isotropic, weightless half-space. Neither knows anything about the soil beyond that. What they show is a consequence of geometry alone, which is why it is worth looking at before any constitutive model is chosen.
Take m in both cases, the circle being 3.0 m in diameter:
| Circle | Strip | |
|---|---|---|
| 0.5 | 0.6464 | 0.8183 |
| 1.0 | 0.2845 | 0.5498 |
| 1.5 | 0.1462 | 0.3958 |
| 2.0 | 0.0869 | 0.3058 |
| 3.0 | 0.0403 | 0.2084 |
| 4.0 | 0.0230 | 0.1575 |
At one width down, the circle has shed 72% of the applied pressure and the strip 45%. At four widths the circle is down to 2.3% and the strip is still at 15.75%.
Where the usual rules of thumb come fromPermalink to “Where the usual rules of thumb come from”
The convention that “significant” stress ends where the increment falls to 20%, or to 10%, of the applied pressure gives very different depths for the two shapes:
| Threshold | Circle | Strip | Ratio |
|---|---|---|---|
| 1.25 B | 3.13 B | 2.51 | |
| 1.85 B | 6.34 B | 3.42 |
The familiar “two times the width” rule is a rule for pad footings. Applied to an embankment or a long raft it understates the depth of influence by a factor between 2.5 and 3.4, and the error is in the unsafe direction for settlement: a compressible layer sitting at 4B is negligible under a pad and carries a fifth of the applied pressure under a strip of the same width.
The reason is dimensional. A circular load sheds stress into an area growing as ; a strip sheds it into a width growing as . The strip has one fewer direction to spread into, and the difference compounds with depth rather than being a fixed offset.
What this decides in a modelPermalink to “What this decides in a model”
Mesh depth. A boundary placed at 2B below an axisymmetric footing is defensible. The same boundary under a plane-strain embankment is inside the loaded zone, and a fixed boundary there will return settlements that are too small, by an amount that looks like soil stiffness and is actually the mesh.
Which layers matter. Site investigation is routinely terminated at a depth chosen from the pad-footing rule. For a plane-strain problem, the stratum that governs settlement may be below the deepest borehole, and no amount of care in the analysis recovers information that was never collected.
Which idealisation to use when neither fits. A rectangular raft is neither a strip nor a circle. Modelled as plane strain it will be too stiff in the out-of-plane direction and its settlements too large; modelled axisymmetric with an equivalent radius it will shed stress too fast and its settlements too small. The two idealisations bracket the answer, which is more useful than either alone — running both and reporting the range is an honest treatment of a problem that two dimensions cannot represent.
The one that catches peoplePermalink to “The one that catches people”
A strip footing on a genuinely infinite elastic half-space has infinite settlement. The integral of the strain over depth does not converge, because the stress decays as and there is no datum. This is not a numerical artefact: it is a property of the solution.
Finite settlements in plane strain come from one of three things — a rigid stratum at finite depth, stiffness increasing with depth, or a settlement quoted relative to a point rather than to infinity. Every practical plane-strain settlement calculation contains one of them, usually implicitly. When a plane-strain model returns settlements that grow as the mesh is made deeper, that is not a convergence failure to be fixed by refinement; it is the model answering the question that was asked.
Which is a good illustration of a general point about verification: a result that keeps changing under refinement is sometimes converging on the right answer to the wrong problem. Distinguishing the two requires knowing the closed form, which is why benchmarks against analytical solutions are the foundation of a verification suite rather than an ornament on it — and why a published benchmark reproduced to 0.03% is worth the effort of setting up.