Thick rectangular plate bending — transverse shear, uniform load

• ShellMITC4 — production flat-plate slab element
• ElasticMembranePlateSection (Mindlin/FSDT plate)
• Transverse-shear correction κ = 5/6
• Hard simply-supported, uniform pressure
• Thickness sweep a/t = 50 → 5 (6 cases)
• Exact FSDT Navier series (closed form)
• h-refinement convergence to closed form

A thick rectangular plate carries a uniform out-of-plane pressure and its center deflection is compared to the exact first-order shear-deformation (FSDT / Reissner–Mindlin) Navier series. This is the companion to SB5: where SB5 keeps the plate thin (a/t = 200) so transverse shear vanishes, SB6 makes it thick (a/t from 50 down to 5) so transverse-shear deflection becomes a large part of the total (2–17% here) and the Mindlin element must capture it. Following CSI SAP2000 Verification Example 2-012 (Plate Bending with Shear Deformation; Roark & Young), six configurations are run — a square (1:1) and a 2:1 plate, each at several thicknesses, all hard simply-supported under uniform pressure. STRIX recovers the scale-invariant coefficient α = w·D/(q·a⁴) and compares it to the series value. The reference uses the same continuum theory the element discretizes (FSDT with κ = 5/6), so the comparison is like-for-like and near-exact; the shear bracket [1 + D·k²/(κGh)] tends to 1 as the plate thins, recovering the SB5 / Timoshenko value. The plate is a production flat-plate slab, meshed with the exact ShellMITC4 + ElasticMembranePlateSection element a real STRIX slab uses.

1. α — 1:1, a/t=10w = α·q·a⁴/D, FSDT κ=5/6 (shear 4.9%)= 0.004273
2. α — 1:1, a/t=5w = α·q·a⁴/D, FSDT κ=5/6 (shear 17.2%)= 0.004904

Exact first-order shear-deformation (FSDT / Reissner-Mindlin) Navier series for a hard simply-supported rectangular plate under uniform pressure (Wang relationship, exact for SS plates): w_mn = Q_mn/(D·k⁴)·[1 + D·k²/(κGh)], κ = 5/6, ν = 0.3. The eight-case-style layout and thick-plate/shear focus follow CSI SAP2000 Verification Example 2-012 (Plate Bending with Shear Deformation; Roark & Young). STRIX recovers the scale-invariant deflection coefficient α = w·D/(q·a⁴), a = short side; the shear bracket → 1 as a/t → ∞ (recovers the SB5/Timoshenko thin-plate value) and grows with thickness.

For every case the first-order MITC4 plate approaches the FSDT coefficient monotonically from below (a coarse mesh is slightly stiff), reaching the closed form to well under a tenth of a percent at the finest mesh. Because the reference is the exact FSDT solution with the element’s own shear correction (κ = 5/6), the match is near-exact across the whole thin-to-thick range — the recovered coefficient rises from the thin-plate value at a/t = 50 to a strongly shear-amplified value at a/t = 5, tracking the analytical shear term throughout. §7 shows the governing (worst final) case; all six appear at the finest mesh in §6. The auto rotational stabiliser is stripped and its measured effect here is under 0.1%.

1. Exact first-order shear-deformation (FSDT / Reissner–Mindlin) Navier series for a hard simply-supported plate under uniform load: w_mn = Q_mn/(D·k⁴)·[1 + D·k²/(κGh)], κ = 5/6, ν = 0.3 (Wang relationship, exact for SS plates); see §5.
2. R. J. Roark & W. C. Young, Formulas for Stress and Strain — flat plates with shear deformation; the thick-plate/shear focus follows CSI SAP2000 Software Verification Example 2-012 (Plate Bending with Shear Deformation).
3. STRIX headless harness: production flat_plate slab → generatePyForLoadCase → opensees.pyd; ShellMITC4 + ElasticMembranePlateSection (κ = 5/6 confirmed to ±0.06% vs FE), hard simply-support, auto rotational stabiliser removed.

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SB6.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SB6/

sha256

(pending publish)