Rigid diaphragm torsion
The distribution of a story shear to the walls and frames below a diaphragm that is not flexible, by ASCE/SEI 7-22 12.8.4: the center of mass and the center of rigidity, the inherent and accidental torsion, the torsional irregularity ratio TIR and Type 1 of Table 12.3-1, the amplification Ax, and the direct and torsional shear of every element in both plan directions. It is an analysis: the result reads Calculated.
Inputs of the rigid diaphragm torsion #
- Level and story shear: the plan dimensions Lx and Ly (coordinates are measured from the lower-left corner; the plan edges x = 0, Lx and y = 0, Ly are where the torsional irregularity ratio is taken and 5 % of the dimension perpendicular to the force is the accidental eccentricity) and the strength-level story shears Vx and Vy below the diaphragm (from the seismic base shear calculation; 0 skips a direction). The rigid diaphragm basis records how the diaphragm is idealized: a concrete slab or concrete-filled deck meeting 12.3.1.2 (span-to-depth ratio 3 or less, no horizontal irregularity of Types 2 to 5: confirmed by you), or a semirigid diaphragm you idealize as rigid (12.3.2.1.1 and 12.8.4.3 permit the rigid assumption for TIR and Ax).
- Walls and frames: each element with its name, the direction of the force it resists (x or y) and its center in plan. Its rigidity is either entered (kip/in, or relative values used consistently) or computed from the wall length L, thickness t, the story height h and the modulus E as a concrete or masonry pier, cantilever k = E t / (4 (h/L)³ + 3 h/L) or fixed at both ends k = E t / ((h/L)³ + 3 h/L) (flexure and shear, G = 0.4 E).
- Center of mass: point weights (the diaphragm, walls, equipment at their centroids) and rectangular area pieces of uniform weight (an L- or T-shaped floor), or the centroid of the rectangular plan for a uniform mass.
- Torsion (code options): the seismic design category; whether more than 75 % of the story's lateral strength is at or on one side of the center of mass (the second Type 1 condition of Table 12.3-1, from the element strengths: yours); whether accidental torsion is included in the design forces where 12.8.4.2.1 requires it (prefilled) or always; Ax from this level's rigid-diaphragm displacements or from δmax / δavg entered from your building model; and the option that a torsional shear never reduces the direct shear (an office practice, not an ASCE 7 requirement: off by default).
Checks performed #
| Check | Basis |
|---|---|
| Rigidity of the elements | Entered, or pier flexure + shear; Σ kx, Σ ky |
| Center of mass | Σ w x / W, Σ w y / W of the weights and area pieces (or the plan centroid) |
| Center of rigidity, J | x_cr = Σ ky x / Σ ky, y_cr = Σ kx y / Σ kx; J = Σ kx (y − y_cr)² + Σ ky (x − x_cr)² over the elements of both directions |
| Torsional irregularity ratio | TIR = Δmax / Δavg (Eq. 12.3-2) at the two plan edges, from the rigid-body drift V/Σk ± (T/J) × distance, with accidental torsion each way and Ax = 1 (12.3.2.1.1); the larger of the two directions |
| Torsional irregularity | Table 12.3-1 Type 1: TIR > 1.2 or the 75 % strength condition; the requirements of Table 12.3-1a (12.7.3, 12.5.4, 12.8.6, 12.3.3.5, 12.2.5.4, 12.3.4.2.1, 12.9.1.5) are listed; an advisory, not a failure |
| Accidental torsion | 12.8.4.2.1: in the design forces for SDC B with Type 1 and TIR > 1.4, and SDC C to F with Type 1 (or always, your option); 5 % of the plan dimension perpendicular to the force each way (12.8.4.2.2) |
| Amplification Ax | Eq. 12.8-15, Ax = (δmax / 1.2 δavg)², 1.0 to 3.0, SDC C to F with Type 1 (12.8.4.3) |
| Element forces | Direct shear V k / Σk (parallel elements) plus torsional shear T k d / J (every element), T = V (e ± Ax e_a); each element takes the larger of the accidental torsion cases with the sign of its torsional shear kept (C12.8.4.2), and the larger of the two directions |
| Redundancy | TIR > 1.4 in both directions in SDC D to F: ρ = 1.3 (12.3.4.2.1), an advisory |
The drawing (on screen, in the report and as a PDF / DXF sheet) is the level plan with the elements as entered (walls with their length and thickness, elements entered by rigidity as short bars), each tagged with its name and design shear (or its name only when there are more than six; the sheet notes list every shear), the center of mass and the center of rigidity, the story shears on their line of action and the plan dimensions. A plan too large for the smallest plan scale is compressed and drawn not to scale.
Limitations #
- One level at a time: the story shear and the rigidities are this level's. Ax by Eq. 12.8-15 uses displacements relative to the base; this level's own rigid-body displacements are exact for a one-story structure and are the level's drift otherwise: enter δmax / δavg from your building model for a multistory structure.
- Elements parallel to x or y only (no skewed walls); no flexible or semirigid diaphragm analysis: the diaphragm is idealized as rigid.
- The 100 % + 30 % orthogonal combination (12.5) is not formed; the forces are per direction. Under concurrent orthogonal loading 12.8.4.2.2 permits the accidental offset in one direction only.
- The 75 % strength condition of Type 1 is your input (element strengths are not part of the calculation).
- The forces are QE (strength level, before ρ and the load combinations); story drift limits (12.12) and the design of the walls, frames, chords and collectors are separate calculations.
- Table 12.3-1a cites 12.3.3.4 for the collector amplification while the body text is 12.3.3.5, and lists 12.8.4.2 under SDC D to F while 12.8.4.2.1 says SDC C to F; the body text is followed (to confirm against the code text: ASCE 7-22 errata).
Verification examples #
Verification status: 62 values compared in 10 examples (7 published, 3 hand calculations from the cited equations): 62 within tolerance, 0 recorded deviations. Code basis: the provisions it uses were read in the target edition.
Published worked examples (research/notes/05_lateral_rigid_diaphragm_torsion.md):
| Example | What it checks | Published | ILX Calcs |
|---|---|---|---|
| YAW-CM-CR | L. Yaw, Rigid Diaphragm Analysis (Walla Walla University, 2022): one-story concrete wall building, 80 × 50 ft | x_cm 40.592, y_cm 26.016 ft; relative rigidities 1.2898, 1.1119, 0.3754; x_cr 37.037, y_cr 38.729 ft; J 4,548.7 | same within 0.01 % |
| YAW-VX | Force along x, 82.14 k | direct 63.62 / 18.52 k, inherent ∓3.34 / ±10.97 k, accidental 0.66 / 2.16 k, design 60.94 / 22.51 / 13.12 k | same within 0.01 % |
| YAW-VY | Force along y, 82.14 k | direct 44.11 / 38.03 k, inherent ∓3.07 k, accidental 3.45 / 1.05 k, design 44.50 / 44.55 / 1.98 k | same within 0.01 % |
| SEAOC-DE33 | SEAOC 2018 IBC Seismic Design Manual Vol. 1, Design Example 33 (via the free Tekla Tedds verification page), e + e_a = 24 ft | CR (20, 20) ft, e_a 4 ft, direct 75 / 25 k, totals 52.5 / 47.5 k | same (the code envelope of the ± e_a cases gives wall A 60.0 k) |
| FEMA-13.2-B | FEMA P-1051 §13.2, 16 equal masonry walls on plank, accidental torsion only | Σd² 11,520, Mta 7.6 V, wall D 0.125 V + 0.0238 V = 0.149 V | same within 0.2 % |
| FEMA-13.2-D | Same building in SDC D: δmax / δavg at the plan end, Ax | 1.402, Ax 1.365, wall D 67.4 kips | 1.401, 1.363, 67.2 kips (FEMA rounds the ratio first) |
| FEMA-11.2 | FEMA P-1051 §11.2, precast building, four transverse walls | 95 + 14.25 = 109.25 kips | same |
Hand calculations: HC-1 (core walls: J = 400, TIR 2.25, Ax capped at 3.0, shears 87.5 / 37.5 k), HC-2 (eccentric center of rigidity: x_cr = 20 ft, J = 3,200, envelope shears 52.125 / 44.625 / 7.3125 k, TIR 1.326) and HC-3 (L-shaped floor: x_cm 35.714, y_cm 18.571 ft, W 280 k), all exact.