FERS vs the standard NAFEMS benchmarks

Natural frequencies of a pin-jointed planar '+' cross with distributed member mass — the single-cross sibling of FV2, with the same frequencies at multiplicity 1/3/1/3.

Element:

1D beam-frame (Euler/Timoshenko)

Material:

E = 200 GPa, ν = 0.3, ρ = 8000 kg/m³

Geometry:

Four 5 m arms radiating from a rigid centre; solid 0.125 × 0.125 m square section.

Supports:

Four outer tips pinned; out-of-plane DOFs restrained (in-plane spectrum).

Loading:

None — eigenvalue (normal-modes) analysis.

Targets derived in closed form (paywalled in NAFEMS R0015) and cross-checked against the FV2 double-cross theory values.

Natural frequencies of a pin-jointed planar frame with distributed member mass, including repeated (degenerate) eigenvalues.

Element:

1D beam-frame (Euler/Timoshenko)

Material:

E = 200 GPa, ν = 0.3, ρ = 8000 kg/m³

Geometry:

Eight 5 m arms radiating at 45° from a rigid centre (four 10 m beams crossing at midpoint). Solid 0.125 × 0.125 m square section.

Supports:

Eight outer tips pinned; out-of-plane DOFs restrained to isolate the in-plane spectrum.

Loading:

None — eigenvalue (normal-modes) analysis.

The 7-fold degenerate clusters show a sub-0.15% discretisation split — an honest finite-element artefact, well within benchmark tolerance.

A completely unsupported (free-free) frame — 3 rigid-body modes plus the elastic in-plane spectrum. Exercises rigid-body-mode handling.

Element:

1D beam-frame, free-free

Material:

E = 200 GPa, ν = 0.29, ρ = 8000 kg/m³

Geometry:

Closed 10 × 10 m square of four rigid-jointed 10 m beams; solid 0.25 × 0.25 m square section.

Supports:

Completely free (free-free); out-of-plane DOFs restrained for the in-plane family.

Loading:

None — eigenvalue analysis (3 in-plane rigid-body modes at ≈0 Hz).

Solved free-free directly — the modal solver applies an automatic spectral shift for the singular (rigid-body) stiffness, returning the 3 rigid-body modes at ≈0 Hz. Values are the converged Euler-Bernoulli frequencies, which are slightly more accurate than NAFEMS's coarse-mesh reference at the higher modes.

The marquee deep-beam dynamic benchmark: shear deformation and rotary inertia across bending, torsional and extensional mode families.

Element:

1D beam (Timoshenko: shear + rotary inertia)

Material:

E = 200 GPa, ν = 0.3, ρ = 8000 kg/m³, shear coefficient κ = 5/6

Geometry:

Length 10 m, solid circular section (Ø ≈ 2.28 m, L/D ≈ 4.4). Radius fitted to the flexural fundamental; torsional & extensional modes then confirm the geometry independently.

Supports:

End A: uₓ=u_y=u_z=θₓ=0. End B: u_y=u_z=0 (simply supported in bending, free axially/torsionally).

Loading:

None — eigenvalue (normal-modes) analysis.

Torsional and extensional modes match to <0.1%; the higher flexural modes carry ~2–3% finite-element dispersion — comparable to or better than commercial codes' own published FV5 results (e.g. DIANA 1.9% / 6.2%).

Flexural vibration of a complete free-free ring, both in-plane and out-of-plane (flexural-torsional), for circumferential wavenumbers n = 2, 3, 4.

Element:

1D curved beam (polygonised), free-free

Material:

E = 200 GPa, ν = 0.3, ρ = 8000 kg/m³

Geometry:

Ring of centroidal radius 1.0 m, solid circular section Ø 0.10 m, modelled as 60 straight beam segments.

Supports:

Completely free (free-free); the modal solver's spectral shift handles the singular stiffness, returning 6 rigid-body modes at ≈0 Hz.

Loading:

None — eigenvalue analysis.

Reference values are the classical thin-ring closed form (targets in R0015 are paywalled). The residual ~1% is the genuine difference between a straight-segment ring and curved-ring theory; FERS also lands within ~1.8% of the NAFEMS R0015 finite-element values.

Transverse-bending natural frequencies of a thin square plate — exercises the plate/shell modal mass added in engine 0.2.43 (before which plate DOFs were massless and had no modes).

Element:

2D flat plate bending (Mindlin/DSG3 shell)

Material:

E = 210 GPa, ν = 0.3, ρ = 7850 kg/m³

Geometry:

Square plate, side a = 1 m, thickness t = 0.01 m (a/t = 100, thin); 20×20 mesh (800 triangular DSG3 elements).

Supports:

Simply supported (w = 0) on all four edges; in-plane and drilling DOFs fixed to isolate the transverse-bending spectrum.

Loading:

None — eigenvalue (normal-modes) analysis.

Not a numbered NAFEMS FVxx case (R0015's plate frequencies are paywalled) — validated instead against the classical Navier closed form f_mn = ½·π·(m²+n²)·√(D/ρt)/a². The low-order Mindlin element converges from above; the fundamental lands at 1.49% (20×20), higher modes carry more low-order dispersion.

In-plane membrane stress on a curved-boundary quarter model — the canonical low-order membrane convergence benchmark.

Element:

2D flat membrane (plane-stress CST)

Material:

E = 210 GPa, ν = 0.3

Geometry:

Quarter elliptic annulus: inner ellipse a=2.0/b=1.0 m, outer a=3.25/b=2.75 m, thickness 0.1 m.

Supports:

Symmetry: uₓ=0 on the y-axis edge, u_y=0 on the x-axis edge.

Loading:

Uniform outward normal traction of 10 MPa on the outer elliptic edge.

Mesh convergence: 12×3: 24.0% → 24×4: 14.6% → 48×6: 6.4% → 72×8: 2.7%

Low-order (CST) membrane stress converges from below toward the target as the mesh refines; the shipped model (72×8) lands within 2.7%.

Axial warping stress in an open thin-walled Z-section under end torque — a benchmark the reference solves with a folded shell, and that a standard 6-DOF frame solver physically cannot represent.

Element:

1D thin-walled warping beam (7-DOF, Vlasov)

Material:

E = 210 GPa, ν = 0.3

Geometry:

Open Z-section (web 2 m, two 1 m flanges, wall t=0.1 m), length 10 m. Warping constant C_w = 0.0417 m⁶; shear centre at the centroid (point-symmetric).

Supports:

Fully clamped including the warping DOF at the fixed end.

Loading:

End torque of 1.2 MN·m about the beam axis.

FERS's 7-DOF beam carries the St-Venant + restrained-warping (Vlasov) torsion, and the warping normal stress σ = bimoment·ω_n/C_w reproduces the shell target to 0.3% — a capability most frame solvers lack.

Plate bending on a sharply-skewed 30° acute rhombus with a moment singularity at the obtuse corners — a severe low-order plate test.

Element:

2D flat plate bending (Mindlin)

Material:

E = 210 GPa, ν = 0.3

Geometry:

Rhombus, all sides 1.0 m, 30° acute / 150° obtuse, thickness 0.01 m (t/L = 0.01).

Supports:

Simply supported (w = 0) on all four edges; rotations free.

Loading:

Uniform normal pressure 0.7 kPa.

Mesh convergence: 8×8: 50.6% → 16×16: 27.9% → 24×24: 14.4% → 32×32: 7.5% → 48×48: 2.0%

The in-browser model is 32×32 (7.5%, within the free browser tier) and the Live column reproduces it exactly; refining offline to 48×48 reaches 2.0%. Convergence is monotonic toward the target.