DcrPMMHinge3d custom P-M-M column hinge — element-mechanics defense

• DcrPMMHinge3d — DCR self-authored OpenSees plugin (no third-party example exists)
• Isolated single-hinge rig (same strategy as the plugin’s own regression gates)
• Uniaxial FEMA-356/ASCE-41 4-branch backbone — exact per-branch algebra
• Elastic unload/reload secant
• Biaxial Bresler-type capacity-envelope projection (DCR’s own §5.9 algorithm)
• N-dependent capacity via independently re-implemented Fritsch-Carlson PCHIP

DcrPMMHinge3d is a zero-length, 6-DOF P-M-M column hinge that DCR authored and compiled as an OpenSees runtime plugin — there is no CSI/NAFEMS published example to compare against (unlike SP1), and the element has no bare-static-load-case production path either (it only appears inside the Pushover/THA RC-column FEMA lumped-hinge n-method assembly, which drags in a full structure and an arc-length driver — exactly the global-response complexity that made SP1 hard). SH1 instead isolates the ELEMENT ITSELF: two coincident nodes, one hinge, driven directly by DisplacementControl / prescribed-displacement ramps — the same isolation strategy the plugin author’s own regression gates use (gate_p1.tcl/gate_p2.tcl) — and compares it against closed-form expectations re-derived from scratch in this benchmark, never by reading the plugin’s C++ source. Honesty split (devplan §12.5/§12.11 precedent): quantities A/B/D verify the element against a published physical model — the standard FEMA-356/ASCE-41 four-branch generalized backbone (elastic→hardening→post-capping softening→residual) and the Fritsch & Carlson (1980) monotone-cubic interpolation algorithm — independently re-coded, not copied. Quantities C (biaxial Bresler projection) instead verify the implementation against DCR’s own §5.9 design specification (independent per-direction backbones + closed-form radial capacity projection) — the interaction surface FORM is a published power-law/Bresler-type envelope, but this specific combination algorithm has no external authority to validate against, since no other software runs this code.

1. Uniaxial backbone, cap boundary (kappa=thetaP)Rc*Mu= 1.25×10⁸ N·mm
2. Elastic unload from mid-hardening (theta1->theta2, Delta_theta=-0.005 rad)k0*(theta2-thetaP_committed), thetaP_committed=theta1-M1/k0= 6.25×10⁷ N·mm
3. Biaxial Bresler projection, symmetric ratio (kappaZ=kappaY=0.010)beta*Mhz, beta=min(1,1/Phi), Phi=(|Mhz/Ccapz|^a+|Mhy/Ccapy|^a)^(1/a)= 7.0871×10⁷ N·mm
4. N-dependent cap via PCHIP, N=-1.5e6 N (interior query, table N=[-2e6,-1e6,0,1e6])Rc*pchip(Mzu_table, N)= 9.125×10⁷ N·mm

No third-party published example exists for this DCR-original element (unlike SP1/CSI 1-026) — devplan §12.11 honesty split: (a) the uniaxial FEMA-356/ASCE-41 4-branch backbone (elastic/hardening/post-capping softening/residual) and (b) the Fritsch & Carlson (1980, SIAM J. Numer. Anal. 17(2):238-246) monotone-cubic capacity interpolation are PUBLISHED external models — re-derived from scratch in benchmarks/nonlinear/SH1_pmmHingeBackbone.ts, never by reading DcrPMMHinge3d.cpp. (c) The biaxial Bresler-type capacity-envelope projection (independent per-direction backbones + closed-form radial scale-back beta=min(1,1/Phi)) is DCR's own §5.9 design choice — the interaction SURFACE FORM (normalized power-law) is standard/published, but this specific combination algorithm has no external authority to validate against; quantities C_* verify the implementation is self-consistent with DCR's own stated specification, not that the specification itself reproduces a third-party benchmark.

SH1 drives a single, non-refinable zero-length element (a bare hinge has no mesh to converge) — each row instead summarizes the worst deviation across that family’s checkpoints. A (uniaxial backbone) exercises all four branches in one continuous monotonic push; B checks the elastic unload secant from a plastic state; C checks the biaxial Bresler projection at two load ratios; D checks the N-dependent capacity at two interior PCHIP query points on a 4-level table (the plugin author’s own gate_p2.tcl explicitly flags PCHIP as untested in Tcl — its 2-level table degenerates PCHIP to linear interpolation; SH1’s 4-level table has genuine curvature at both queries). All four families land at machine precision (worst diff on the order of 1e-10%), because the closed form is not an approximation of the production algorithm here — on a piecewise-linear-in-kappa backbone the plastic-multiplier root is unique, so the independently re-derived algebra IS the same root the production cutting-plane Newton solve converges to.

1. FEMA 356 / ASCE 41-13 — generalized force-deformation backbone for deformation-controlled actions (4-branch: elastic→hardening→post-capping softening→residual plateau).
2. Fritsch, F.N. & Carlson, R.E. (1980) — Monotone Piecewise Cubic Interpolation, SIAM Journal on Numerical Analysis 17(2):238–246 (PCHIP tangent formula, independently re-implemented in sh1PchipTangents/sh1PchipEval).
3. Bresler, B. (1960) — reciprocal-load biaxial bending interaction (normalized power-law interaction surface form; the specific decoupled-backbone + radial-projection combination algorithm is DCR’s own §5.9 design, not from Bresler).
4. opensees_plugins/DcrPMMHinge3d/ — DcrPMMHinge3d.cpp (element under test), gate_p1.tcl/gate_p2.tcl (plugin author’s own regression gates, isolation-rig pattern reused here), dcr_pmm_plugin_element_devplan.md.

Engine

v1.0.6 (tcl)

Run date

2026-07-11

Record

records/SH1.json

Evidence archive

verif-evidence-eng1.0.2-win-x64.zip · SH1/

sha256

(pending publish)