Response-spectrum analysis of a three-dimensional L-shaped braced frame

• 3-D response-spectrum analysis (RSA)
• Braces / axial-only members (linear Truss)
• Rigid floor diaphragms (Ux, Uy, Rz)
• Rotational mass (MMI) — offset-mass condensation
• Tabulated El-Centro spectrum · 5% damping
• CQC · SRSS · ABS · axial-force recovery

SR2 verified the 3-D response-spectrum pipeline on a moment frame; SR2b takes it onto the brace surface and adds rotational mass. Adopted as CSI SAP2000 Verification Example 1-025 with the independent solution from Peterson (1981), a three-story, L-shaped building is framed by four identical two-bay X-braced frames whose columns and diagonals carry axial force only — every member is emitted as a linear element Truss. All mass is concentrated at a centre-of-mass joint per level with X and Y translational mass and a rotational mass moment of inertia (MMI) about Z; the plan is L-shaped, so the CM is offset from the centre of rigidity and the response is translation–torsion coupled. The excitation is the El-Centro-derived tabulated response spectrum in X with 5% damping, combined by CQC, SRSS and ABS; only the first two modes drive the RSA. The production mass path injects translational mass only, so the MMI is reproduced physically by four offset translational masses on the diaphragm — the de-Zhu T-matrix mass condensation (M_red = TᵀMT) turns them into the exact Z rotational inertia, with no production-code change. The reference is reproduced directly at its own discretisation (84 truss members, 3 concentrated story masses).

1. Frequency — mode 1(K − ω²M)·φ = 0, f = ω/2π= 3.0592 Hz
2. Roof CM Ux — CQCu = √(Σᵢ Σⱼ ρᵢⱼ·uᵢ·uⱼ), ρ = Der Kiureghian= 1.0329 in
3. Axial elm 1 — CQCN = √(Σᵢ Σⱼ ρᵢⱼ·Nᵢ·Nⱼ)= 279.48 kip

CSI SAP2000 Software Verification, Example 1-025 (Response-Spectrum Analysis of a Three-Dimensional Braced Frame); independent solution = Peterson (1981), reproduced exactly (0 %) by SAP2000. A three-story, L-shaped building framed by four identical two-bay, three-story X-braced planar frames whose columns and diagonals carry axial force only (all pinned). The frames are tied by a rigid floor diaphragm at each level. All mass is concentrated at a centre-of-mass joint per level with X and Y translational mass (1.24224 kip-s²/in) and a rotational mass moment of inertia about Z (174,907.4 kip-in-s²), giving nine dynamic DOF; for consistency with Peterson only the first two modes drive the RSA. The excitation is the El-Centro-derived 5%-damped response spectrum applied in X, combined by CQC, SRSS and ABS. Reference values are the two modal frequencies, the roof centre-of-mass (joint 51) X/Y displacement and Z rotation, and the axial forces in three Frame-1 elements (a column and two diagonals) under each combination rule.

The reference publishes the roof centre-of-mass displacement under three modal-combination rules, and STRIX reproduces each: CQC and SRSS through the production RsaCalculator (Der Kiureghian CQC), and ABS (absolute sum) from the same per-mode responses. Because the two modes are closely spaced (frequencies within 2 %), CQC (1.0329 in) far exceeds SRSS (0.7372 in) — the cross-correlation adds the modes nearly in phase in X — while ABS (1.0423 in) is the upper bound. All three land within tolerance of the independent reference, verifying that STRIX handles closely-spaced modes correctly (the very case that motivates CQC).

1. CSI SAP2000 Software Verification, Example 1-025 — Response-Spectrum Analysis of a Three-Dimensional Braced Frame.
2. F. E. Peterson (1981) — independent response-spectrum solution reproduced exactly by SAP2000.
3. E. L. Wilson, A. Der Kiureghian & E. P. Bayo, EESD 9(2), 1981 — CQC rule. STRIX harness: buildRsaPy → opensees.pyd (de-Zhu T-matrix, M_red = TᵀMT + Transformation) → rsa_modal.bin; MMI via four offset translational masses on the rigid diaphragm.

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SR2b.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SR2b/

sha256

(pending publish)