Concrete stair
A straight waist-slab stair, with or without landings, designed per 12 in. strip as a one-way slab to ACI 318-19 Chapter 7 with the ASCE 7-22 2.3.1 load combinations (strength design). The stair is analyzed on its horizontal projection: under vertical loads the bending moment of the kinked slab equals that of the projection, the shear normal to the flight is V cos θ, and the vertical deflection of the inclined waist is that of a member of stiffness Ec Ie cos θ on the projection.
Inputs of the concrete stair #
- Materials: f'c, Grade 60 or 80 bars, normalweight or lightweight concrete, Ec by 19.2.2.1 (as the other concrete types).
- Stair: riser R and tread T, the number of risers (the flight has one tread fewer: the top landing is the last step), the waist thickness t measured perpendicular to the soffit from the inside corners of the steps, and the supports: at the outer ends of the landings (bottom and top landing lengths, 0 = none, and the landing thickness), or at the ends of the flight (landings carried by beams or walls of their own). Each end is pinned (simple bearing) or fixed / continuous (built into a wall or beam that restrains rotation: top bars are then designed). The support width sets the critical section for shear at d from its face (7.4.3.2). The stair width is used for the drawings; the design is per 12 in. strip. Clear cover: 3/4 in. prefilled (Table 20.5.1.3.1).
- Reinforcement: bottom (main) bars and spacing, top bars and spacing at fixed ends, distribution bars across the stair (24.4.3).
- Loads: area loads on plan per load type (finishes, live load; ASCE 7-22 Table 4.3-1: 100 psf for stairs, 40 psf in one- and two-family dwellings) and the self-weight: γ(t √(R² + T²)/T + R/2) on the flight (the waist and the triangular steps) and γ tl on the landings.
- Deflection: span / n limits for the immediate live load deflection and the deflection after attachment (Table 24.2.2), the load duration (ξ) and the sustained part of the live load: the engineer's.
Auto-design (editor toolbar) varies the waist thickness (1/2-in. steps; the landings follow when they had the same thickness) and the bottom, top and distribution bars (size and spacing). Objectives: lowest cost index (concrete, reinforcement); thinnest waist, then cost; least reinforcement. The steps, landing lengths, supports and loads stay the engineer's; see Auto-design.
Checks of the concrete stair #
| Check | Basis |
|---|---|
| Flexure, bottom bars | the largest positive Mu in the flight and in each landing against φMn of that part's thickness by strain compatibility (22.2, φ from Table 21.2.2), per 12 in. |
| Flexure, top bars | the negative Mu at each fixed / continuous end (7.5.2.1, 22.2) |
| Tension-controlled section | εt ≥ εty + 0.003 (7.3.3.1, as printed) |
| Minimum reinforcement | As ≥ 0.0018 b h per part (7.6.1.1) |
| Bar spacing | ≤ min(3h, 18 in.) (7.7.2.3) and the crack-control spacing of Table 24.3.2 with fs = 2/3 fy (24.3.2.1) |
| Distribution bars | ≥ 0.0018 Ag of the thicker part, spacing ≤ min(5h, 18 in.) (24.4.3.2, 24.4.3.3) |
| One-way shear | Vu (normal to the flight in the flight) ≤ φVc with Vc = 8λsλ(ρw)^(1/3)√f'c b d (Table 22.5.5.1(c)), at d from the support face (7.4.3.2), no stirrups (7.6.3.1) |
| Minimum thickness | Table 7.3.1.1 on the horizontal span (ℓ/20, ℓ/24, ℓ/28 by the end conditions, × (0.4 + fy/100,000)): a reference only; the deflections are always calculated |
| Deflections | Ie of Table 24.2.3.5 for the waist at the largest positive service moment (24.2.3.7), stiffness Ec Ie cos θ over the whole span; immediate ΔL and λΔ(ΔD + sustained ΔL) + ΔL (24.2.4.1) against the entered limits (Table 24.2.2) |
| Notes | re-entrant corners (bars cross at the top of the flight and are developed past the kink), the 300 lb tread load of ASCE 7-22 Table 4.3-1, ℓd of the bottom bars (25.4.2.4) |
Limitations #
- Straight flights spanning in their own direction only; stairs spanning across the width (between stringers or walls), cantilevered treads, free-standing (dog-leg without support, helical) stairs and stairs with intermediate supports are not covered.
- The moments come from an elastic analysis of the projection with a uniform stiffness; for fixed ends the actual distribution varies somewhat with the different thicknesses and the slope of the flight.
- The deflection uses the flight's stiffness (Ec Ie cos θ) over the whole span, conservative where the landings are as thick or thicker; Ie is taken at the largest positive service moment.
- The axial force in the flight is neglected (it is small and compressive near the bottom support).
- The bar lengths, the extension past each kink, the top-bar extent, hooks and laps, and the supporting walls or beams are not designed.
- No free published worked example of an ACI stair is on disk; the module is verified by hand calculations (research/DOCUMENTS_NEEDED.md).
Verification examples #
Verification status: 11 values compared in 3 examples (0 published, 3 hand calculations from the cited equations): 11 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: partial. Only some provisions were read in the target edition; the rest rests on older editions, examples or summaries.
Hand calculations in closed form (backend/tests/test_verify_concrete_stair.py): R = 7 in., T = 11 in., 10 risers (flight run 8.25 ft), waist 8 in., f'c = 4,000 psi, Grade 60, #5 at 6 in. bottom, D = 15 psf finishes plus the self-weight, L = 100 psf. The flexure, minimum steel and effective moment of inertia used are those of the one-way slab, which is verified against the StructurePoint one-way slab example (DE-OWS).
| Example | What it checks | Hand calculation | ILX Calcs |
|---|---|---|---|
| ST-1 | Flight only, pinned ends | flight self-weight 162.3 psf on plan, Mu = 3.171 kip-ft, φMn = 18.08 kip-ft, Vu = 1.011 kips (V cos θ at d from the face), φVc = 6.169 kips, ΔL = 0.0067 in. (uncracked, Ec Ig cos θ) | equal |
| ST-2 | Flight with two 3 ft landings, pinned at the landing ends | Mu at midspan = 9.125 kip-ft | equal |
| ST-3 | Flight only, fixed at both ends | Mu− = wu ℓ²/12 = 2.114 kip-ft, φMn of #4 at 12 in. = 6.168 kip-ft, Mu+ = wu ℓ²/24 = 1.057 kip-ft | equal |