Modal response spectrum analysis
The modal response spectrum analysis of ASCE/SEI 7-22 12.9.1 on a planar stick model: one weight per level and one shear stiffness per story. The calculation finds the natural periods and mode shapes, the participation factors and the effective modal masses. It combines the modal responses by SRSS or CQC and scales the forces to the ELF base shear (12.9.1.4). It reports the level forces, story shears, overturning moments and drifts, and checks the design story drift against Table 12.12-1. It is an analysis: the result reads Calculated.
Inputs of the modal analysis #
- Design spectrum: the multi-period design spectrum of 11.4.5.1, entered as Sa = 2/3 of the USGS multi-period MCER values at the 22 periods (Seismic site parameters gives them). The sample is FEMA P-2192 Table 3-2 × 2/3, so enter your site's values. The two-period spectrum (11.4.5.2, from SDS, SD1 and TL) is offered for where Exception 2 of 11.4.5 applies.
- SDS, SD1, S1, TL: used for the ELF base shear of 12.9.1.4 (Eqs. 12.8-2 to 12.8-7), Cu and the seismic design category.
- Risk category: Ie from Table 1.5-2.
- System: a Table 12.2-1 system (R, Ω0, Cd, the Ta structure type) or the values entered. The approximate period is Ct hn^x or 0.1N. The redundancy factor ρ is used for Δa/ρ.
- Stick model: for each level, its height above the base, its seismic weight wx (12.7.2) and the lateral shear stiffness of the story below it (story shear over story drift, from a model of the structure). Up to 60 levels.
- Modes: all modes (100 % of the mass, 12.9.1.1), or the fewest that reach 90 % (the exception). This is your choice.
- Combination: CQC (the default) or SRSS (12.9.1.3). For CQC, the modal damping is 5 % by default, the damping of the 11.4 spectra (C12.9.1.3).
- Allowable drift Δa / hsx: the Table 12.12-1 coefficient, which you enter.
Checks performed #
| Check | Basis |
|---|---|
| Natural modes | K φ = ω² M φ (shear building, lumped masses), periods, Γ, effective modal mass, cumulative participation (12.9.1.1) |
| Modal responses | Sa(T) of 11.4.5 divided by R/Ie (12.9.1.2); displacements and drifts × Cd/Ie |
| Combination | SRSS or CQC with ρij = 8ζ²(1 + r) r^1.5 / ((1 − r²)² + 4ζ² r (1 + r)²), each quantity combined on its own (12.9.1.3) |
| ELF base shear | T = min(T1, Cu Ta) (12.9.1.4.1); Cs by Method 1 (multi-period) or Method 2 (two-period) with Eqs. 12.8-6 and 12.8-7; V = Cs W |
| Scaling | Forces × V/Vt where Vt < V (12.9.1.4.1); drifts × Cs W/Vt where Eq. 12.8-7 governs and Vt < Cs W (12.9.1.4.2) |
| Story drift | Δ = Cd Δe / Ie × the drift scale, against Δa (Table 12.12-1; Δa/ρ for moment frames in SDC D to F, 12.12.1.1) |
| 3-D model | 12.9.1.8 requires a 3-D model: stated as an advisory |
The drawings show the stick elevation with the combined, scaled level forces, and the story shear and overturning moment diagrams.
Limitations #
- Planar model. The model is planar, one direction at a time, with a fixed base and shear-building stories. 12.9.1.8 requires a 3-D model, so use this for preliminary design, to check a 3-D model, or where you judge that it represents the structure.
- Not covered. Accidental torsion (12.9.1.5), P-delta (12.9.1.6; Story drift takes these results), soil-structure interaction (12.9.1.7), orthogonal effects (12.5), CQC-4, and the residual rigid mode of 12.9.1.1 (a stick model has all its modes).
- Combined values do not balance. Level forces, shears and moments are each combined separately, so they are not in equilibrium.
- No auto-design. Nothing is sized.
Verification examples #
Verification status: 26 values compared in 5 examples (0 published, 5 hand calculations from the cited equations): 26 within tolerance, 0 recorded deviations. No published example: every value is a hand calculation. That checks the program against this reading of the code, not the reading itself. Code basis: the provisions it uses were read in the target edition.
No published stick-model example is on disk. The FEMA P-1051 and P-2192 modal examples are 3-D models from analysis programs. The values are therefore checked against closed forms:
| Example | What it checks | Value |
|---|---|---|
| Two equal masses and stiffnesses | ω² = (k/m)(3 ∓ √5)/2, Γ1, effective masses 94.72 % and 5.28 %, ρ12, SRSS and CQC base shears, ELF V and V/Vt | T1 1.0166 s, T2 0.3883 s, Vt (CQC) 54.24 kips |
| Uniform five-story building | ωj = 2√(k/m) sin((2j − 1)π/22) | T1 2.2075 s, T2 0.7563 s, T3 0.4797 s |
| One story in the plateau | Vt = W SDS/R, δ = Cd Sa g/(R ω²) | 48.26 kips, 0.2654 in. |
| S1 = 0.8 g | Cs by Eq. 12.8-7, drift scale Cs W/Vt | 1.910 |