Elastic Link (twoNodeLink) Beta Angle coordinate transformation — inclined 6-DOF spring

• twoNodeLink (Elastic Link) — 6 generalized DOF springs, non-coincident nodes
• Beta-Angle → -orient conversion ("MIDAS convention", TclBuilder.writeElasticLinks) — DCR-authored, never independently checked before
• Both up-vector branches exercised (near-vertical vs general member)
• Anisotropic 6-DOF stiffness + general 6-component load (no symmetry to hide an axis error)
• Tcl engine only — Elastic Link is absent from StaticPyBuilder

STRIX exposes a "Beta Angle" field for Elastic Links (the MIDAS-style member-orientation convention), converted at runtime into the -orient vector pair OpenSees's twoNodeLink element expects. This conversion is DCR's own code — never independently verified — unlike the element formulation itself (stock OpenSees, trusted the same way ASDShellQ4's own formulation is trusted). No CSI/NAFEMS published example fits this scope (CSI's Group 6 Link catalog is entirely dynamic/nonlinear — ramp loading, gap, hook, damper, isolator — with no purely static linear-spring test), so this benchmark is a self-designed closed form: a fully fixed ground node connected to one free node by a single, inclined Elastic Link with six distinct (anisotropic) DOF stiffnesses, loaded by a fully general 6-component point load. Because every stiffness direction is distinct, any sign flip, axis swap, or wrong up-vector branch in the Beta-Angle conversion changes every displacement component — a symmetric/isotropic transverse stiffness would hide such an error entirely, since rotating an isotropic spring about its own axis changes nothing. Two geometry variants exercise both branches of the up-vector selection ("near-vertical member → up = Global X" vs "otherwise → up = Global Z"). De-risk finding: an initial closed-form draft assumed the six directions were fully independent springs; a controlled single-component-load experiment against the real engine revealed a pure moment load producing a nonzero translation response. Fetching OpenSees's own TwoNodeLink.cpp source confirmed this is real, documented element kinematics for a non-coincident link (shear-direction deformation includes a lever-arm term from both nodes' rotations, located at the -shearDist fraction of the element length) — not DCR's own code, so it was folded into the closed form rather than treated as a defect, and the corrected closed form matches the real engine to machine precision.

1. Oblique Uxu = ((C*T)^T diag(k_local) (C*T))^-1 F, T from MIDAS Beta-Angle frame (betaDeg=25), C = TwoNodeLink shear-rotation coupling (shearDist=0.5)= 0.152961 mm
2. NearVert Uxu = ((C*T)^T diag(k_local) (C*T))^-1 F, T from MIDAS Beta-Angle frame (betaDeg=-40), C = TwoNodeLink shear-rotation coupling (shearDist=0.5)= −0.12171 mm

Self-designed closed form (no CSI/NAFEMS published example fits — CSI Group 6 Link catalog is dynamic-only, see devplan §12.3 SB12 note). Independent reference = exact linear-algebra congruence transform K_global = (C*T)^T * diag(k_local) * (C*T), solved by an independent Gaussian-elimination linear solve. T = blockdiag(R,R), R = MIDAS Beta-Angle local frame re-derived from scratch (benchmarks/element/SB12_elasticLink.ts, no import from production TclBuilder code) -- this is the transform under test. C = OpenSees TwoNodeLink's own shear/rotation basic-deformation coupling (TwoNodeLink.cpp setTranLocalBasic(), confirmed against OpenSees/OpenSees GitHub source; shearDist fixed 0.5 as DCR always emits) -- real, documented, non-DCR element physics, included only so the closed form matches the engine, not itself under test. Not a FEM discretization/convergence problem -- exact linear algebra, so near-machine-precision agreement with the simulated result is expected.

Not a mesh-convergence study — this is exact linear algebra (a 6×6 congruence transform + direct solve), not a discretized FEM problem, so no mesh exists to refine. The two rows instead demonstrate branch coverage: the oblique variant exercises the "otherwise → up = Global Z" branch of the Beta-Angle convention, while the near-vertical variant exercises the opposite "near-vertical member → up = Global X" branch — the two mutually-exclusive code paths inside TclBuilder.writeElasticLinks. Both reproduce the independent closed form to under 0.0003% across all twelve probed quantities (6 DOF × 2 variants), consistent with the expected near-machine-precision agreement of an exact linear system.

1. OpenSees — TwoNodeLink element (Wiki): https://opensees.berkeley.edu/wiki/index.php/Two_Node_Link_Element
2. OpenSees — TwoNodeLink.cpp source (setTranLocalBasic()), OpenSees/OpenSees GitHub — basic-deformation shear/rotation coupling formula.
3. MIDAS — Elastic Link (Beta Angle member-orientation convention): https://support.midasuser.com/hc/en-us/articles/17917025888281-Elastic-Link
4. CSI — SAP2000 Analysis Verification, Group 6 (Link) catalog — surveyed and found to contain no static linear-spring example (all dynamic/nonlinear), motivating this self-designed closed form.

Engine

v1.0.6 (tcl)

Run date

2026-07-11

Record

records/SB12.json

Evidence archive

verif-evidence-eng1.0.2-win-x64.zip · SB12/

sha256

(pending publish)