SB1 · STATIC-BEAM · Element Benchmarks

Euler-Bernoulli 1D cantilever (tip deflection)

Tip deflection under a transverse point load — 1-D flexural absolute check

PASS +0.00020% error tolerance 1% −0.107865 mm vs −0.107865 mm
§2

Technical features tested

  • elasticBeamColumn — linear-static beam element
  • Euler–Bernoulli flexure (no shear area emitted)
  • Strong-axis routing $Iy ← sec.Iz
  • geomTransf Linear, auto vecxz [0,0,1]
  • Nodal load & single-node fixity
§3

Problem description

A prismatic cantilever of length L is fully fixed at node N1 and free at node N2, where a downward transverse load P acts along global −Z. The tip deflection is compared with the Euler–Bernoulli closed-form solution. Because the static element carries no shear area, the response is pure bending — the element should reproduce the closed form to machine precision, independent of subdivision.

N1 N2 P = 1 kN δ L = 3000 mm Z X
Figure SB1. Cantilever geometry, loading, and deflected shape (deflection exaggerated).
§4

Geometry, properties & loading

Geometry

Length L
3000 mm
Section b×h
300×500
Area A
150000
I strong (Iz)
3.125×10⁹
I weak (Iy)
1.125×10⁹

Material

Model
Elastic
E
26700 MPa
ν
0.2
Units
N · mm

Loading & BC

N1
fixed 6-DOF
N2 load
Fz −1000 N
Self-weight
off
Analysis
Linear static
§5

Reference solution

  1. Tip deflection u_z-P*L^3/(3*E*I)= −0.107865 mm
  2. Tip rotation r_y+P*L^2/(2*E*I)= 5.3933×10⁻⁵ rad

Timoshenko & Gere, Mechanics of Materials — cantilever tip deflection under an end load (Euler-Bernoulli).

§6

Results comparison

Response quantityProbeSTRIXReferenceΔVerdict
Tip deflection u_z (mm) N2.uz −0.107865 −0.107865 +0.00020% PASS
Tip rotation r_y (rad) N2.ry 5.3933×10⁻⁵ 5.3933×10⁻⁵ 0.00% PASS
Support reaction R_z (N) N1.Fz 1000 1000 0.00% PASS
Support moment M_y (N·mm) N1.My −3×10⁶ −3×10⁶ 0.00% PASS
§7

Convergence

Meshuz (mm)Δ
1 element−0.107865 +0.00020%

The cubic Hermite shape functions of the Euler–Bernoulli element are exact for a linearly varying moment field, so a single element already recovers the closed form. Subdivision is mesh-independent — no refinement study is required for this benchmark.

§8

Conclusion

PASS

STRIX reproduces the reference solution to +0.00020% (tolerance 1%). The tested element and its formulation are verified against the reference.

§9

References & analysis files

  1. Timoshenko, S. & Gere, J. — Mechanics of Materials. Cantilever end-load deflection δ = PL³/3EI.
  2. STRIX headless harness: generatePyForLoadCase → python.exe → parseAll.
Engine
v1.0.6 (opensees.pyd)
Run date
2026-08-19
Record
records/SB1.json
Evidence archive
verif-evidence-eng1.0.6-win-x64.zip · SB1/
sha256
(pending publish)