SB5 · STATIC-PLATE · Element Benchmarks

Rectangular plate bending — thin plate, uniform & central point load

Plate bending — production ShellMITC4 slab vs Timoshenko thin-plate theory

PASS −0.621% error tolerance 2% 0.00406 vs 0.00406
§2

Technical features tested

  • ShellMITC4 — production flat-plate slab element
  • ElasticMembranePlateSection (Mindlin plate)
  • Simply-supported & clamped edges
  • Uniform pressure + central point load
  • 1:1 and 5:1 aspect ratios (8 cases)
  • h-refinement convergence to closed form
§3

Problem description

A rectangular plate carries an out-of-plane load and its center deflection is compared to the Timoshenko–Woinowsky thin-plate series solution. Following CSI SAP2000 Verification Example 2-005 (geometry after MacNeal–Harder), eight configurations are run — a square (1:1) and a 5:1 rectangular plate, each simply-supported and clamped, under a uniform pressure and a central point load. STRIX recovers the dimensionless deflection coefficient α = w·D/(q·a⁴) (uniform) or w·D/(P·a²) (point), which is scale-invariant and compared to the published table value. The plate is a production flat-plate slab, meshed with the exact ShellMITC4 + ElasticMembranePlateSection element a real STRIX slab uses. It is kept thin (a/t = 200) so transverse shear is negligible and this Mindlin element converges to the thin-plate coefficient (the thick-plate regime is covered separately). At structural scale the automatic rotational stabiliser is negligible and is removed.

q P w (center) a SS or clamped 1:1 & 5:1
Figure SB5. Rectangular plate (1:1 shown; 5:1 also tested) — supported edges, uniform pressure q and central point load P, monitored by the center deflection w.
§4

Geometry, properties & loading

Geometry

Width a
2000 mm
Aspects
1:1 & 5:1
Thickness t
10 mm
a / t
200

Material

Model
Elastic · thin plate
E
30000 MPa
ν
0.3
Units
N · mm

Loading & BC

Uniform q
1×10⁻⁴ N/mm²
Point P
100 N
Supports
SS & clamped
Probe
center defl. w
§5

Reference solution

  1. α — SS, 1:1, uniformw = α·q·a⁴/D (Table 8)= 0.00406
  2. α — SS, 1:1, pointw = α·P·a²/D (Eq.147, Table 23)= 0.0116

Timoshenko & Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (1959): Table 8 (SS, uniform), Eq.147 & Table 23 (SS, central point), Table 35 (clamped, uniform), Table 37 (clamped, central point), ν = 0.3. The eight-case layout follows CSI SAP2000 Verification Example 2-005 (geometry after MacNeal & Harder, 1985). STRIX recovers the scale-invariant deflection coefficient α (= w·D/(q·a⁴) uniform, w·D/(P·a²) point) and compares it to the published table value; a = short side.

§6

Results comparison

Response quantityProbeSTRIXReferenceΔVerdict
Deflection coefficient α SS · 1:1 · uniform 0.00406 0.00406 +0.064% PASS
Deflection coefficient α SS · 5:1 · uniform 0.01294 0.01297 −0.264% PASS
Deflection coefficient α Fixed · 1:1 · uniform 0.00126 0.00126 +0.336% PASS
Deflection coefficient α Fixed · 5:1 · uniform 0.00259 0.0026 −0.506% PASS
Deflection coefficient α SS · 1:1 · central point 0.0116 0.0116 −0.0020% PASS
Deflection coefficient α SS · 5:1 · central point 0.01694 0.01695 −0.069% PASS
Deflection coefficient α Fixed · 1:1 · central point 0.00559 0.0056 −0.165% PASS
Deflection coefficient α Fixed · 5:1 · central point 0.00721 0.00725 −0.621% PASS
§7

Convergence

MeshalphaΔ
8×400.00689 −4.944%
16×800.00715 −1.329%
24×1200.00721 −0.621%

For every case the first-order MITC4 plate approaches the Timoshenko coefficient monotonically from below (a coarse mesh is slightly stiff), reaching the closed form to a fraction of a percent at the finest mesh — comfortably better than the commercial verification tolerance for the same problem. §7 shows the governing (worst final) case; all eight appear at the finest mesh in §6. The plate is thin, so the residual transverse-shear of this Mindlin element is under 0.1%; the auto rotational stabiliser is stripped and was measured to shift the answer by under 0.2% even when present.

§8

Conclusion

PASS

Across all eight CSI 2-005 configurations — two aspect ratios, simply-supported and clamped, uniform and central point load — the recovered deflection coefficient matches the Timoshenko–Woinowsky thin-plate solution to well within tolerance, converging monotonically. The production slab element ShellMITC4 + ElasticMembranePlateSection is verified for out-of-plane plate bending under both distributed and concentrated load and both support conditions. Thick-plate (transverse-shear) behaviour is verified separately against Roark.

§9

References & analysis files

  1. S. Timoshenko & S. Woinowsky-Krieger, Theory of Plates and Shells, 2nd ed. (1959) — Tables 8, 23, 35, 37 & Eq.147 (ν = 0.3); see §5.
  2. CSI SAP2000 Software Verification, Example 2-005 — Rectangular Plate with Static Loads (geometry after MacNeal & Harder, 1985); same eight-case layout, center-deflection check.
  3. STRIX headless harness: production flat_plate slab → generatePyForLoadCase → opensees.pyd; ShellMITC4 + ElasticMembranePlateSection, auto rotational stabiliser removed.
Engine
v1.0.6 (opensees.pyd)
Run date
2026-08-19
Record
records/SB5.json
Evidence archive
verif-evidence-eng1.0.6-win-x64.zip · SB5/
sha256
(pending publish)