Response-spectrum analysis of a three-dimensional eccentric rigid-diaphragm frame

• 3-D response-spectrum analysis (RSA)
• Rigid floor diaphragms (Ux, Uy, Rz)
• Eccentric mass — translation–torsion coupling
• SRSS · CQC · ABS · NRC-10%
• Per-mode inertia-force solve + force recovery
• Bending + axial · 4% damping

SR1 verified the response-spectrum pipeline on a planar frame; SR2 takes it to three dimensions with rigid floor diaphragms and a plan mass eccentricity, the configuration real buildings present. Adopted as CSI SAP2000 Verification Example 1-024 with the independent solution from Peterson (1981), a two-story, two-bay-by-two-bay fixed-base moment frame carries its story mass at an eccentric centre of mass (offset from the geometric centre), and is excited by a constant 0.4 g response spectrum in X with 4% damping. The eccentricity couples translation and torsion, so the roof response depends on the full 3-D rigid-diaphragm condensation and the modal-combination rule. It exercises the entire 3-D seismic path: the rigid-diaphragm modal solve (four finite modes from a rank-deficient reduced mass — no rotational inertia, matching the reference), the per-mode inertia-force static analysis under the diaphragm constraint, spectral scaling, and all four combination rules (SRSS, CQC, ABS, NRC-10%). Following the reference, bending and axial deformations are retained and shear is made rigid; the mass acts in X and Y through the production Loads-to-Masses path. The reference is reproduced directly at its own discretisation.

1. Period — mode 1(K − ω²M)·φ = 0, T = 2π/ω= 0.2271 s
2. Roof CM Ux — SRSSu = √(Σ (Γ_m·Sa_m/ω_m²·φ_m)²)= 0.02012 ft
3. Roof CM Ux — NRC 10%u = √(Σ uₘ² + 2 Σ_{|Δf|/f ≤ 10%} |uᵢ·uⱼ|)= 0.02016 ft

CSI SAP2000 Software Verification, Example 1-024 (Response-Spectrum Analysis of a Three-Dimensional Moment Frame); independent solution = Peterson (1981), reproduced exactly by SAP2000. A two-story, two-bay-by-two-bay fixed-base 3-D moment frame with a plan eccentricity between the geometric centre (35, 25 ft) and the centre of mass (38, 27 ft), excited by a constant 0.4 g response spectrum in the X direction with 4% modal damping. Bending and axial deformations are retained; shear is ignored (shear area 0). Story mass acts in X and Y only (no rotational mass inertia), giving four natural modes. Reference values (Peterson) are the four modal periods and the roof centre-of-mass X-displacement under four modal-combination rules (SRSS, CQC, ABS, NRC-10%).

The reference publishes the roof centre-of-mass displacement under four modal-combination rules, and STRIX reproduces each: SRSS and CQC through the production RsaCalculator (Der Kiureghian CQC), and ABS (absolute sum) and NRC-10% (U.S. NRC Regulatory Guide 1.92 ten-percent method) combined from the same per-mode responses. The four rules span a narrow band (SRSS 0.02012 → ABS 0.02050 ft) because the fundamental X mode dominates the eccentric torsional contributions; all four land within tolerance of the independent reference — verifying the complete combination surface, not only the rule the reference happens to favour.

1. CSI SAP2000 Software Verification, Example 1-024 — Response-Spectrum Analysis of a Three-Dimensional Moment Frame.
2. F. E. Peterson (1981) — independent response-spectrum solution reproduced exactly by SAP2000.
3. U.S. Nuclear Regulatory Commission, Regulatory Guide 1.92, Rev. 1 — modal-combination rules (grouping / ten-percent methods). E. L. Wilson, A. Der Kiureghian & E. P. Bayo, EESD 9(2), 1981 — CQC rule. STRIX harness: buildRsaPy → opensees.pyd (de-Zhu T-matrix + Transformation) → rsa_modal.bin.

Engine

v1.0.6 (opensees.pyd)

Run date

2026-08-19

Record

records/SR2.json

Evidence archive

verif-evidence-eng1.0.6-win-x64.zip · SR2/

sha256

(pending publish)