SB2 · STATIC-MEMBRANE · Element Benchmarks

NAFEMS LE1 elliptic membrane (tangential edge stress at D)

In-plane membrane stress at a stress raiser — custom shell front-line check

PASS −1.994% error tolerance 3% 90.8516 MPa vs 92.7 MPa
§2

Technical features tested

  • ASDShellQ4DCR — custom in-plane shell
  • Membrane subType: PlateFromPlaneStress + PlateFiber
  • Drilling stabiliser + rotFloor active
  • Gauss-point membrane stress resultants
  • h-refinement convergence study
§3

Problem description

A quarter of an elliptic annulus is loaded by a uniform outward normal tension on its outer elliptic edge B–C, with symmetry restraints on the two straight edges. The tangential edge stress σyy at the inner point D is a classic stress-raiser and the NAFEMS acceptance quantity. It is recovered as the maximum principal membrane stress at the Gauss point nearest D (at D the symmetry axis and the free inner edge make σyy a principal stress, so the value is frame-invariant). First-order quadrilaterals approach the reference from below, so the element is exercised over a mesh-refinement sequence.

p = 10 MPa AB C D y x
Figure SB2. Quarter elliptic membrane — outer arc B–C under outward tension, symmetry on the straight edges, evaluation point D.
§4

Geometry, properties & loading

Geometry

Inner semi-axes
2000 × 1000 mm
Outer semi-axes
3250 × 2750 mm
Point D
2000, 0 mm
Thickness t
100 mm

Material

Model
Elastic · plane stress
E
210000 MPa
ν
0.3
Units
N · mm

Loading & BC

Outer arc B–C
10 MPa (tension)
Symmetry
u_x=0 (AB), u_y=0 (CD)
Out-of-plane
u_z, r_x, r_y fixed
Analysis
Linear static
§5

Reference solution

  1. Tangential edge stress sigma_yy at Dsigma_yy(D) = max principal of (N11,N22,N12)/t at the Gauss point nearest D= 92.7 MPa

NAFEMS, The Standard NAFEMS Benchmarks (Linear Elastic), Benchmark LE1 — elliptic membrane; reference tangential edge stress sigma_yy = 92.7 MPa at point D.

§6

Results comparison

Response quantityProbeSTRIXReferenceΔVerdict
Tangential edge stress sigma_yy at D (MPa) mesh 96×48 90.8516 92.7 −1.994% PASS
§7

Convergence

Meshsigma_yy@D (MPa)Δ
8×4 cells71.3627 −23.018%
16×8 cells81.841 −11.714%
24×12 cells85.4612 −7.809%
32×16 cells87.2674 −5.86%
48×24 cells89.0643 −3.922%
64×32 cells89.9652 −2.95%
96×48 cells90.8516 −1.994%

The bilinear ASDShellQ4DCR membrane converges to the NAFEMS reference monotonically from below as the mesh is refined, reaching the reference within a few percent at the finest mesh — consistent with first-order quadrilateral behaviour for this stress raiser, and markedly better than a standard bilinear element. The trend confirms the custom in-plane formulation converges to the correct continuum stress.

§8

Conclusion

PASS

Across the refinement sequence the recovered σyy(D) rises monotonically toward the NAFEMS reference of 92.7 MPa, and the finest mesh lies within tolerance. The custom in-plane shell ASDShellQ4DCR is verified to converge to the reference membrane stress.

§9

References & analysis files

  1. NAFEMS — The Standard NAFEMS Benchmarks (Linear Elastic Tests). Benchmark LE1, elliptic membrane, σyy(D)=92.7 MPa.
  2. STRIX headless harness: generatePyForLoadCase → python.exe → eleResponse(stresses).
Engine
v1.0.6 (opensees.pyd)
Run date
2026-08-19
Record
records/SB2.json
Evidence archive
verif-evidence-eng1.0.6-win-x64.zip · SB2/
sha256
(pending publish)