A reinforced concrete stair is a slab that happens to be sloping. Once the geometry is set out and the loads are converted onto plan, the design is the same K, z and As you already know from slab design.
Two things are genuinely different, and they are where stairs go wrong: the self weight has to be worked out on plan when the concrete is on a slope, and stairs are almost always governed by deflection rather than by strength.
Step 1 — Set out the steps
The same three-storey building as the rest of these examples: a 3.0 m storey height, climbed in two flights with a half-landing between them, so each flight rises 1.5 m.
riser R = 150 mm -> 1500 / 150 = 10 risers
going G = 250 mm -> 9 goings = 2250 mm on plan
comfort 2R + G = 2(150) + 250 = 550 mm (550 to 700 is comfortable)
A flight always has one more riser than it has goings — the top riser lands on the landing, which has no tread of its own. Counting that wrong is how a flight ends up 150 mm short of the floor it was meant to reach.
The flight spans from a beam at floor level to the half-landing beam, and carries 750 mm of the landing with it:
waist h = 150 mm, cover 25 mm, T12 main bars
materials fcu 25, fy 460
Step 2 — The load, converted onto plan
This is the step that is unique to stairs. The waist is a slab measured along the slope, but the span, the bending moment and the imposed load are all measured on plan. So the waist's self weight has to be spread over the shorter plan length, which makes it heavier per square metre than its thickness suggests.
Waist 0.15 x 24 x 1.166 = 4.20 kN/m²
Steps 0.5 x 0.15 x 24 = 1.80 kN/m² (triangular)
Finishes terrazzo, plaster, soffit = 1.20 kN/m²
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Gk = 7.20 kN/m²
Qk = 3.0 kN/m² (a shared stair; a stair inside one
dwelling takes the 1.5 of its floor)
n = 1.4(7.20) + 1.6(3.0) = 10.08 + 4.80 = 14.88 kN/m²
The steps themselves are the triangles of concrete sitting on the waist. Averaged over the going they are exactly half a riser deep, which is where 0.5 × R × 24 comes from — the one line of stair loading nobody remembers and everybody can re-derive.
Step 3 — Moment and steel
Per metre width of flight, simply supported:
V = nL/2 = 14.88 x 3.0 / 2 = 22.3 kN/m
d = 150 - 25 cover - 12/2 = 119 mm
K = M / (b d² fcu) = 16.74 x 10&sup6; / (1000 x 119² x 25) = 0.047
z = d[0.5 + sqrt(0.25 - 0.047/0.9)] = 0.944d = 112 mm
As = M / (0.87 fy z) = 16.74 x 10&sup6; / (0.87 x 460 x 112) = 373 mm²/m
Provide T12 @ 250 c/c = 452 mm²/m, in the bottom, running up the flight
Distribution steel across the flight, at the 0.13% minimum:
Provide T10 @ 300 c/c = 262 mm²/m
Step 4 — Deflection, which is what really governs
BS 8110 allows a stair flight a 15% longer span/depth ratio than a plain slab (cl 3.10.2.2), provided the flight itself occupies at least 60% of the span. It does here — 2.25 of 3.0 m is 75%.
M/bd² = 16.74 x 10&sup6; / (1000 x 119²) = 1.18
MF = 0.55 + (477 - 252) / (120(0.9 + 1.18)) = 1.45
allowable = 20 x 1.45 x 1.15 = 33.3
actual = 3000 / 119 = 25.2
25.2 < 33.3 PASS
Shear, for completeness — it never governs a domestic stair, and it is worth seeing by how much:
vc = (0.79/1.25)(0.380)1/3(400/119)1/4 = 0.620 N/mm²
0.188 << 0.620 PASS, by a factor of three
A waist that fails deflection is fixed by thickening it, and a thicker waist is heavier, which raises the moment — so the check has to be re-run, not adjusted in your head. The slab example shows the same loop with a 150 mm slab that only works at 160.
Step 5 — Detailing the bars
The arithmetic is the easy half. What keeps a stair standing is where the steel goes:
- Main bars run up the slope, in the bottom of the waist, and continue into the landing.
- At a kink where the flight meets the landing, tension bars must not be bent around the re-entrant corner — the force in them tries to burst the concrete straight off. Lap them past each other into the opposite face instead. This is the classic stair detailing failure and it is not visible in any calculation.
- Top steel over a supporting beam wherever the flight is continuous, to control the cracking that a simply supported analysis pretends is not there.
- Anchorage into the landing beam of a full tension anchorage length — 40φ in grade 25 concrete.
Structura's calc sheets defer laps and anchorages to detailing and say so on the sheet. That is a boundary, not an omission: a bar bending schedule can be generated from a design, but the kink detail above depends on how the flight actually meets the landing.
How the quantity is expressed
Stairs are measured per metre width of flight, the same way one-way slabs are measured per metre width. A 1.2 m wide flight is 1.2 times what came out above. Multiply once, at the end, and write down which of the two the number is — a stair costed as if it were per flight when it was per metre is a 20% error in the concrete and nobody notices until the block is being cast.
Run a stair yourself
Structura designs stairs to the same method — slope factor, load on plan, bending, the 1.15 stair allowance on span/depth — and prints it as the calc sheet above, with a bar bending schedule and quantities beside it. Single members, stairs included, are free to run, as many as you like.
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