Bathe & Wilson eigenvalue problem — ten-bay nine-storey plane frame

• Eigenvalue (modal) analysis
• Custom sparse solver — scipy eigsh
• ElasticTimoshenkoBeam frame element
• 297-DOF ten-bay nine-storey frame
• First three eigenvalues ω²
• Shear-rigid · reference discretisation

The Bathe & Wilson eigenvalue problem (1972) is the classical large multi-degree-of-freedom frame for dynamic eigenvalue extraction, adopted verbatim as CSI SAP2000 Verification Example 1-021 (and by SeismoStruct, Tekla, OpenBrIM). A ten-bay, nine-storey fixed-base plane frame — all members identical, one element per member — is solved and its first three eigenvalues ω² compared to the published values, exercising STRIX’s custom sparse eigensolver (scipy eigsh) on a large frame. The reference is shear-rigid, so the production ElasticTimoshenkoBeam shear term is made rigid via the section shear-area modifier; the frame is laid in-plane with out-of-plane DOF restrained, giving exactly the reference’s three DOF per node. It is solved at the reference’s own one-element-per-member discretisation.

1. Eigenvalue ω² — mode 1 (1st lateral sway)(K − ω²M)·φ = 0= 0.589541 rad²/s²
2. Eigenvalue ω² — mode 2 (2nd lateral sway)(K − ω²M)·φ = 0= 5.52695 rad²/s²
3. Eigenvalue ω² — mode 3 (3rd lateral sway)(K − ω²M)·φ = 0= 16.5878 rad²/s²

K.-J. Bathe & E. L. Wilson, "Large Eigenvalue Problems in Dynamic Analysis," ASCE J. Eng. Mech. Div. 98(EM6), 1972 (with an independent solution by Peterson, 1981). The canonical multi-DOF frame eigenvalue benchmark, reproduced as CSI SAP2000 Verification Example 1-021, SeismoStruct Verification Ex.10, Tekla Structural Designer and OpenBrIM Frame21. A ten-bay, nine-storey fixed-base 2-D frame; all members identical, one finite element per member, consistent mass from the line mass, shear-rigid (bending + axial only). The published reference values are the first three eigenvalues ω² (rad²/s²).

The published eigenvalues are those of the Bathe-Wilson model at its defining discretisation — one element per member. STRIX reproduces that model and matches all three eigenvalues to under 0.001% — the same order of agreement SAP2000 itself reports for this example. This table is therefore a mesh-sensitivity study, not a convergence to the reference: refining each member (2, 4 elements) moves the lumped-mass eigenvalues toward the continuum, bounding the discretisation effect to under ~1.4% (largest on the third mode; the fundamental shown here stays within a twentieth of a percent). The §6 comparison is taken at the reference’s own discretisation.

1. K.-J. Bathe & E. L. Wilson, “Large Eigenvalue Problems in Dynamic Analysis,” ASCE J. Eng. Mech. Div. 98(EM6), 1972; independent solution by Peterson (1981).
2. CSI SAP2000 Software Verification, Example 1-021 — Bathe and Wilson eigenvalue problem (modal analysis, line mass on frame objects); same model reproduced by SeismoStruct Ex.10, Tekla Structural Designer, OpenBrIM Frame21.
3. STRIX headless harness: buildEigenPy → opensees.pyd (scipy eigsh) → modal_props.txt; ElasticTimoshenkoBeam, shear made rigid, reference one-element-per-member discretisation.

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SM5.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SM5/

sha256

(pending publish)