Rigid-diaphragm eigenvalue condensation — eccentric multi-storey building

• Eigenvalue (modal) analysis
• Custom de-Zhu T-matrix condensation
• Rigid floor diaphragms (Ux,Uy,Rz)
• Translation–torsion coupling (eccentric)
• Independent numpy reduced-system reference
• Dense eigh vs sparse eigsh

SM5 verified STRIX’s sparse eigensolver on a bare frame, where the de-Zhu rigid-diaphragm condensation reduces to the identity. SM5b exercises the condensation itself. In an eccentric multi-storey building — four corner columns of equal area but distinct second moment of area — every floor is a rigid diaphragm: STRIX condenses each floor’s column-top nodes to one master with three DOF (Ux, Uy, Rz) through its de-Zhu T-matrix (K_red = TᵀKT). The stiffness eccentricity offsets the centre of rigidity from the centre of mass, coupling translation and torsion so the diaphragm lever-arm rows (u = U − dy·Θ) are genuinely exercised. The reference is the same reduced eigenproblem assembled independently in numpy and solved with the dense scipy.linalg.eigh — a different solver from STRIX’s sparse eigsh.

1. Eigenvalue ω² — mode 1 (Ux–Uy–Rz coupled)(K_red − ω²M_red)·φ = 0= 2070.117026 rad²/s²

Independent reference: the SAME reduced eigenproblem assembled from first principles as a 3N×3N rigid-diaphragm shear building — per-column lateral springs k = 12EI/L³ at their plan lever arms, per-column torsion GJ/L, and lumped floor masses reduced about a reference point (M_ΘΘ = Σm·r², M_UΘ = −Σm·ay, M_VΘ = +Σm·ax) — solved with the DENSE generalised eigensolver scipy.linalg.eigh, a different solver and a separate implementation from STRIX's sparse eigsh. The rigid-diaphragm kinematics (u = U − dy·Θ, v = V + dx·Θ) follow the standard treatment (A. K. Chopra, Dynamics of Structures — rigid diaphragms with stiffness eccentricity). Since STRIX's condensed system and the numpy reference are the same discrete reduced eigenproblem, agreement is expected across all modes to near machine precision — isolating and verifying the de-Zhu condensation that SM5 left at the identity.

The columns are single elements, so STRIX’s condensed system and the numpy reference are the same discrete reduced eigenproblem — agreement is expected across all modes to near machine precision, and §6 confirms it (worst 0.0001% over the six lowest modes). This table instead sweeps the storey count (3, 5, 7 → reduced DOF 9, 15, 21) and compares the fundamental eigenvalue ω² of STRIX’s de-Zhu reduction to the independent numpy reduction, showing the condensation reproduces the reference across building sizes. The tiny residual is the finite shear-rigid factor (the numpy reference is pure Euler).

1. A. K. Chopra, Dynamics of Structures: Theory and Applications to Earthquake Engineering — one-storey and multistorey systems with rigid floor diaphragms and stiffness eccentricity (translation-torsion coupling).
2. Rigid-diaphragm kinematic condensation (master/slave transformation u = U − dy·Θ, v = V + dx·Θ); STRIX de-Zhu sparse T-matrix reduction K_red = TᵀKT, M_red = TᵀMT solved by scipy eigsh.
3. STRIX headless harness: buildEigenPy → opensees.pyd (de-Zhu T-matrix + eigsh); independent reference reference_reduced.py (3N×3N shear building, dense scipy.linalg.eigh).

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SM5b.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SM5b/

sha256

(pending publish)