Published validation cases · in depth

Four validation cases, input for input.

The PLAXIS 2D Validation Manual, Version 8 (Bentley Systems) publishes analytic references alongside its own finite-element results for a set of canonical geotechnical problems. This page takes four of them — two linear-elastic settlement problems and two plastic collapse problems — and runs them through KATAI 2D exactly as that manual defines them: every geometry dimension, material parameter and boundary condition letter for letter, the input a checked-in .k2d file, the run the public katai command line. For each problem, three numbers are placed side by side: the closed-form reference, the PLAXIS 2D result as published, and the KATAI 2D result.

4manual cases, rebuilt letter for letter
4checked-in .k2d input files
3numbers per case — closed form · published · KATAI
2answer classes — elastic settlement, plastic collapse

The comparison protocol

Stated up front, so the numbers can be read fairly.

The PLAXIS values on this page are published values, quoted unchanged from the manual — PLAXIS was not re-run for this comparison, and the two programs do not share a mesh; each discretises the problem with its own settings. What is identical is the problem: the mechanics, the parameters, and the closed-form reference both programs answer to.

PLAXIS 2D is an independent, established finite-element implementation, and its published validation results are quoted here so the reader can see where such an implementation lands on the same problem. Displacement-type finite elements approach an exact limit load from either side depending on discretisation and formulation choices; deviations of a few percent against an exact collapse load are the expected class of result for any such program.

None of these numbers is a one-off study. All four cases are part of the KATAI verification corpus: test_input_corpus re-solves each checked-in file on every build and asserts the values below inside declared tolerance bands, and each case is a row of the generated verification matrix, with its sources in references.bib.

The record

Three numbers per problem.

Percentages in the PLAXIS and KATAI columns are deviations from the closed form — except for the Gibson case, where the finite-layer geometry makes the published PLAXIS value the like-for-like reference (explained in the case).

Manual § Problem · quantity Closed form PLAXIS 2D (published) KATAI 2D
2.1 Smooth rigid strip footing on elastic soil
footing force at s = 10 mm [kN/m]		 15.15 15.24 (+0.6%) 15.32 (+1.1%)
2.2 Strip load on incompressible Gibson soil
centreline surface settlement [m]		 0.050 (half-space) 0.047 (finite layer) 0.0454 (−3.4% vs PLAXIS)
3.1 Bearing capacity of a smooth rigid circular footing
limit pressure [kPa]		 225.6 (slip-line, exact) 220.0 (−2.5%) 233.9 (+3.7%)
3.2 Strip footing on clay with strength increasing with depth
limit pressure [kPa]		 7.80 7.86 (+0.8%) 7.91 (+1.4%)

Case 1 · Validation Manual §2.1

Smooth rigid strip footing on elastic soil.

Giroud (1972) analytic solution for a smooth rigid strip footing on an elastic layer, as published in Section 2.1 of the manual. The manual publishes 15.24 kN/m (+0.6%) for PLAXIS 2D.

The problem, verbatim

KinematicsPlane strain, linear elastic, weightless
StiffnessG = 500 kPa, ν = 1/3 (the .k2d stores E = 2(1+ν)G = 1333.33 kPa)
Footinghalf-width B = 1 m, smooth and rigid: prescribed uy = −10 mm, ux free
Domainhalf-model, 7 m wide × 4 m deep, left edge the symmetry axis
Mesh0.5 m 15-node triangles, from the file's own settings
Phasesweightless gravity initial (an exact nil by construction), then the prescribed-displacement phase
F = 2(1 + ν) G B s / ρ, ρ = 0.88 ⟹ F = 15.15 kN/m at s = 10 mm

Input: tests/corpus/kv-fnd-012-giroud-rigid-footing.k2d

The run

> katai validate tests/corpus/kv-fnd-012-giroud-rigid-footing.k2d
OK: ... satisfies the input contract (0 warning(s))

> katai solve tests/corpus/kv-fnd-012-giroud-rigid-footing.k2d
phase 1/2: Initial phase
phase 2/2: Indent
phase 1/2: ok  max|u| = 0.000000e+00 m
phase 2/2: ok  max|u| = 1.008461e-02 m
solved 2 phase(s) in 0.29 s
F [kN/m]vs analytic
Giroud (1972), analytic15.15—
PLAXIS 2D (published, same manual)15.24+0.6%
KATAI 2D (file's own mesh)15.32+1.1% (+0.5% vs the PLAXIS number)

Reading it. The half-model carries half the footing, so the footing force is F = 2|ΣRy| over the footing nodes, from the support reactions the results carry at the prescribed DOFs (the discrete internal force BTσ, no recovery smoothing). Both programs land slightly above the analytic force, and 0.5% apart from each other: the analytic solution is for an idealised layer; a truncated finite-element domain is marginally stiffer, and the stress concentration at the edge of a rigid punch is resolved only to mesh resolution — both effects push the computed force up, in both programs. The suite additionally asserts that the weightless initial phase displaces by exactly zero and that a footing node carries exactly the imposed 10 mm.

Case 2 · Validation Manual §2.2

Strip load on incompressible Gibson soil.

Gibson (1967) closed form for a strip load on an incompressible non-homogeneous half-space whose stiffness grows linearly with depth, as published in Section 2.2 of the manual. The manual models a 4 m finite layer and publishes 0.047 m for PLAXIS 2D — below the half-space value, as a finite layer must be.

The problem, verbatim

KinematicsPlane strain, weightless
StiffnessE(z) = 299 z kPa measured down from the surface (in the .k2d: E ≈ 0, Einc = 299 kPa/m, yref at the surface)
Poissonν = 0.495 (incompressible in the limit)
Loadstrip load q = 10 kPa over a 1 m half-width
Domainhalf-model, 7 m wide × 4 m deep, left edge the symmetry axis
Mesh0.15 m 15-node triangles, from the file's own settings
s = q / (2 dG/dz) = 0.050 m (half-space; the surface settlement under the load is uniform)

Input: tests/corpus/kv-fnd-011-gibson-strip-load.k2d

The run

> katai validate tests/corpus/kv-fnd-011-gibson-strip-load.k2d
OK: ... satisfies the input contract (0 warning(s))

> katai solve tests/corpus/kv-fnd-011-gibson-strip-load.k2d
phase 1/2: Initial phase
phase 2/2: Strip load
phase 1/2: ok  max|u| = 0.000000e+00 m
phase 2/2: ok  max|u| = 8.838396e-02 m
solved 2 phase(s) in 5.15 s
s [m]
Gibson (1967), half-space closed form0.050upper reference
PLAXIS 2D (published, 4 m finite layer)0.047−6% vs half-space
KATAI 2D (same 4 m layer)0.0454−3.4% vs the PLAXIS number

Reading it. The like-for-like reference here is the published PLAXIS number, because both programs model the same 4 m layer while the closed form is for a half-space; both correctly sit below it, sharing the finite-layer bias. The larger max|u| in the transcript sits at the load edge, where the surface stiffness tends to zero and the load terminates in a local singularity; the verified quantity is the centreline settlement, where Gibson's solution applies. The case is also a discretisation stress test: at ν = 0.495 a low-order displacement element locks volumetrically, and a stiffness that vanishes at the surface amplifies any such error. The 15-node (quartic) triangles carry both without special treatment, landing within 3.4% of the published finite-layer value.

Case 3 · Validation Manual §3.1

Bearing capacity of a smooth rigid circular footing.

Cox (1962) slip-line solution for the indentation of a ponderable c–φ soil, as published in Section 3.1 of the manual: limit pressure pmax = 141 c = 225.6 kPa. The manual publishes 220.0 kPa (−2.5%) for PLAXIS 2D.

The problem, verbatim

KinematicsAxisymmetric Mohr–Coulomb soil cylinder, 5 m radius × 4 m deep
Footingradius R = 1 m, smooth and rigid: prescribed uy = −0.35 m, ux free (smooth contact)
MaterialE = 2400 kPa, ν = 0.20, c = 1.6 kPa, φ = 30°, γ = 16 kN/m³, K0 = 0.5
Flow ruleassociated (ψ = φ = 30°) — a slip-line solution is the associated limit load, so a non-associated run would mix a modelling difference into a verification number
Tension cut-offnone — the slip-line solution has none
Mesh0.25 m 15-node triangles, from the file's own settings
PhasesK0 geostatic initial, then the staged indentation — the soft soil needs that large a punch to reach the collapse plateau
pmax = 141 c = 225.6 kPa

Input: tests/corpus/kv-fnd-013-cox-circular-footing.k2d

The run

> katai validate tests/corpus/kv-fnd-013-cox-circular-footing.k2d
OK: ... satisfies the input contract (0 warning(s))

> katai solve tests/corpus/kv-fnd-013-cox-circular-footing.k2d
phase 1/2: Initial phase
phase 2/2: Indent
phase 1/2: ok  max|u| = 0.000000e+00 m
phase 2/2: ok  max|u| = 4.845403e+00 m
solved 2 phase(s) in 168.00 s
pmax [kPa]vs slip-line
Cox (1962), slip-line (exact)225.6—
PLAXIS 2D (published, same manual)220.0−2.5%
KATAI 2D (file's own mesh)233.9+3.7%

Reading it. Axisymmetric nodal forces are per radian, so the footing force is 2π|ΣRy| and the average pressure p = 2|ΣRy| / R². The two finite-element programs bracket the exact slip-line value from opposite sides — PLAXIS 2.5% below, KATAI 3.7% above — which is precisely how displacement-type formulations behave around an exact collapse load: the sign and size of the bias are properties of the discretisation, not of the mechanics. The case's own mesh study records the sensitivity: a 0.5 m mesh puts only two elements across the footing radius and over-predicts by about +9% (the classic coarse-mesh bearing bias); the file's 0.25 m mesh brings that to +3.7%. The suite additionally asserts the K0 initial phase displaces by exactly 0.000e+00 and a footing node carries exactly the imposed uy = −0.35 m.

Case 4 · Validation Manual §3.2

Strip footing on clay with strength increasing with depth.

Davis & Booker (1973) analytic solution for a smooth strip footing on clay whose undrained strength grows linearly with depth, as published in Section 3.2 of the manual. The manual publishes 7.86 kPa (+0.8%) for PLAXIS 2D.

The problem, verbatim

KinematicsPlane strain, Tresca (φ = 0), weightless
Strengthc(z) = 1 + 2z kPa (c0 = 1 kPa, cinc = 2 kPa/m)
StiffnessE(z) = 299 + 498z kPa, ν = 0.3
Footinghalf-width 1 m (B = 2 m), smooth and rigid, pushed 30 mm
Schemathe two depth profiles are the material's c_inc / E_inc / y_ref fields — this case pins exactly that machinery against a theoretical limit load
Mesh0.5 m 15-node triangles; half-model
pmax = ρ [(2 + π) c0 + B cinc/4] = 7.80 kPa

Input: tests/corpus/kv-fnd-014-davis-booker-strip-footing.k2d

The run

> katai validate tests/corpus/kv-fnd-014-davis-booker-strip-footing.k2d
OK: ... satisfies the input contract (0 warning(s))

> katai solve tests/corpus/kv-fnd-014-davis-booker-strip-footing.k2d
phase 1/2: Initial phase
phase 2/2: Indent
phase 1/2: ok  max|u| = 0.000000e+00 m
phase 2/2: ok  max|u| = 3.804946e-02 m
solved 2 phase(s) in 3.68 s
pmax [kPa]vs analytic
Davis & Booker (1973), analytic7.80—
PLAXIS 2D (published, same manual)7.86+0.8%
KATAI 2D (file's own mesh)7.91+1.4% (+0.6% vs the PLAXIS number)

Reading it. The half-model reads p = |ΣRy| / (B/2) from the reaction output at the prescribed settlement. Both programs land within 1.5% above the analytic collapse load, 0.6% apart from each other, on a problem that couples plasticity to a depth-varying strength and stiffness profile. The suite additionally asserts the weightless initial phase displaces by exactly zero, a footing node carries exactly the imposed 30 mm, and the limit pressure sits within 5% of the analytic value and within 3% of the published PLAXIS number.

Reading the comparison as a whole

What these four cases do and do not say.

On the elastic problems the programs agree with each other to well under one percent where the geometry is like-for-like (Giroud: 0.5% apart), and KATAI sits within 3.4% of the published PLAXIS value on the finite Gibson layer, both codes correctly below the half-space closed form.

On the collapse problems both programs land within a few percent of the exact values — on the same side for Davis & Booker, on opposite sides for Cox. That pattern is worth stating plainly: against an exact limit load, a few percent of discretisation-dependent bias, of either sign, is the honest result class for a displacement finite-element method, and neither program's number should be read as more than that.

Scope. Four problems verify four specific answer classes — elastic settlement (homogeneous and non-homogeneous incompressible), and collapse loads (axisymmetric c–φ and heterogeneous Tresca). No broader claim is made from them; the full verification record lists what else is covered, case by case. Solve timings in the transcripts are from an ordinary 4-core desktop and vary with hardware; the solved numbers do not — the result contract is byte-identical across the front ends (CLI, Python) and across the solver-backend compositions.

Where everything lives

Every artefact, addressable.

The inputs, the write-up and the asserting test are all in the open repository — the record is reproducible from the public tree alone.

ArtefactPath in the repository
The comparison document (this page's source of record, with reading recipes) docs/validation/plaxis-2d-validation-comparison.md
§2.1 Giroud input tests/corpus/kv-fnd-012-giroud-rigid-footing.k2d
§2.2 Gibson input tests/corpus/kv-fnd-011-gibson-strip-load.k2d
§3.1 Cox input tests/corpus/kv-fnd-013-cox-circular-footing.k2d
§3.2 Davis & Booker input tests/corpus/kv-fnd-014-davis-booker-strip-footing.k2d
The asserting test (corpus cases KV-FND-011 … 014, bands declared inline) tests/test_input_corpus.cpp
The generated verification matrix and bibliography docs/validation/verification-matrix.md · docs/validation/references.bib

References

The sources.

  1. PLAXIS 2D Validation Manual, Version 8. Bentley Systems. Sections 2.1, 2.2, 3.1 and 3.2 — the source of the published PLAXIS 2D values quoted on this page, and of the problem statements and analytic reference values as stated there. PLAXIS is a product of Bentley Systems; it is cited here as the published record of an independent finite-element implementation on the same problems.
  2. Giroud, J.-P. (1972). Tables pour le calcul des fondations, Vol. 1: Tassement. Dunod, Paris — the §2.1 reference.
  3. Gibson, R. E. (1967). Some results concerning displacements and stresses in a non-homogeneous elastic half-space. Géotechnique 17(1):58–67, doi:10.1680/geot.1967.17.1.58 — the §2.2 reference.
  4. Cox, A. D. (1962). Axially-symmetric plastic deformation in soils — II. Indentation of ponderable soils. International Journal of Mechanical Sciences 4(5):371–380, doi:10.1016/S0020-7403(62)80024-1 — the §3.1 reference.
  5. Davis, E. H. & Booker, J. R. (1973). The effect of increasing strength with depth on the bearing capacity of clays. Géotechnique 23(4):551–563, doi:10.1680/geot.1973.23.4.551 — the §3.2 reference.

Rerun all four. Today.

Any KATAI 2D install — the single-file katai.exe or the Python wheel — reproduces every run on this page with katai validate and katai solve on the checked-in files. No other tooling.

One caveat, stated rather than left to be discovered: the results above are what the suite pins and what you will reproduce — the displacements are identical to the last digit across builds. The wall-clock times are not. They were measured on one desktop CPU with the v0.8.0 release binary and its vendored Eigen backend, on an otherwise idle machine, and they will differ on yours; nothing in the suite asserts them.

Get KATAI 2D on GitHub Back to the verification record