Staircases

How a reinforced concrete stair is designed with the STATIX staircase calculator on its Eurocode 2 route. It covers the flight and its landings as a one-way slab: the effective span, the actions on plan, and bending, shear and span/d of the waist. It then covers the details: the steel rules of 9.3, the bars at the kinks and the re-entrant corners, cover and fire, and dog-leg stairs on landings that span across.

1. Learning outcomes

After this lesson you can:

treat a waist flight and its landings as a one-way slab (5.3.1, 9.3) and set its effective span (5.3.2.2), with landings in line or landings that span across;
work the actions on plan from the waist, the steps and the finishes, with the imposed load of EN 1991-1-1 and the combination of EN 1990 6.10;
check bending (6.1), shear (6.2.2) and span/d (7.4.2) of the waist, and give the reactions to the supports;
choose the main and distribution bars (9.3.1.1) and detail the kinks and re-entrant corners (8.3, 8.4, 8.7);
check the cover (4.4.1) and the fire resistance (EN 1992-1-2 Table 5.8);
design a landing that spans across a dog-leg stair, and say which checks on the page follow EN 1992-1-1, which are the page’s own choices, and which you do by hand.

2. Before you start

RC Design 1 (Eurocode): bending (lesson 1), deflection and cracking (lesson 4), slabs (lesson 6) and detailing (lesson 8).
The staircase calculator, set to Eurocode 2 (EN 1992-1-1), and its guide.
A pocket calculator and about 60 minutes.
The same lesson on the SA code: RC Design 2, lesson 5 (SANS 10100-1).

What the staircase page’s Eurocode 2 route uses.
The model. EN 1992-1-1 has no stair clause. The page designs a waist flight with its landings as one simply supported one-way slab on the horizontal span, per metre of width. For the span it keeps the stair rule of SANS 10100-1 4.9: the span runs to the centre of a wall bearing, or to half the width of a monolithic beam or wall or of a landing that spans across (at most 900 mm). It keeps two more rules of 4.9: the loads are taken on plan, and the waist is measured square to the soffit. These spans are never shorter than 5.3.2.2 gives (section 3). Straight flights and dog-leg flights onto landings are designed. A flight cantilevered from a wall or spanning between stringers reads NOT CHECKED on this route; cantilever steps and helical stairs are not modelled.
Actions. EN 1990 6.10 with 1.35 G + 1.5 Q (the page does not offer 6.10a and 6.10b); concrete at 25 kN/m³ (EN 1991-1-1 Table A.1); q_k as you give it (EN 1991-1-1 Table 6.2); ψ₂ for the quasi-permanent load (EN 1990 Table A1.1, under More inputs).
Materials. f_ck (cylinder, 12 to 50 MPa) and f_yk (450 or 500 MPa); γ_c 1.5, γ_s 1.15. α_cc and α_ct are 1.0, the recommended values of 3.1.6, with no input, as on the pages of lessons 2 to 4. The course otherwise uses α_cc 0.85 in bending; Example A shows the difference.
Checks. Bending by 6.1 with the rectangular block (λ 0.8, η 1) and x/d ≤ 0.448 (5.5(4) with δ = 1). Shear by 6.2.2(1) at the support lines and at the ends of the flight, with the cap of eq. 6.5, without the β of 6.2.2(6) and without cos θ on the slope. Span/d by 7.4.2, eq. 7.16, with K = 1.0 for every end type, eq. 7.17 capped at 1.5 (the page’s cap) and 7/l_eff over 7 m. A_s,min and A_s,max by 9.3.1.1(1); secondary steel at least 20 % of the main steel (9.3.1.1(2)); spacing by 9.3.1.1(3) with the values for areas of maximum moment everywhere; clear spacing by 8.2(2). Crack control by 7.3.3: up to 200 mm thick the 9.3 rules, thicker by Table 7.2N or 7.3N with the quasi-permanent steel stress. Bottom bars at a wall or beam anchored from its face for F_E = V a_l/z with a_l = d (9.3.1.2(1), 9.2.1.4(2), (3)). Top steel at a monolithic end for 15 % of the span moment, 25 % at a landing across, over 0.2 l (9.3.1.2(2)). Cover by 4.4.1 (c_min,b of the largest main, landing or top bar; c_min,dur of Table 4.4N with class S4; Δc_dev 10 mm). Fire by EN 1992-1-2 Table 5.8 (one-way, simply supported: the axis distance and h_s of the thinnest part).
Landings across. Each is checked as a one-way slab for bending, A_s,min, bar spacing, shear and span/d, for crack control by Table 7.3N when it is over 200 mm thick, and for top steel over its supports when its ends are restrained. Its secondary steel, A_s,max, clear spacing and the anchorage of its bars at the supports have no row (the schedule gives the anchorage length; check that it fits the support). It carries its own load over the whole landing and the flight reaction over the flight width. The page puts the two flights at the very ends of its span and loads the area shared with each flight in both spans (Example B). Its effective span is an input.
Not checked: the mandrel at the upper kinks, the anchorage of the crossing bars at a re-entrant top corner, the point load Q_k of Table 6.2, vibration, and the walls and beams that carry the stair. At the lower soffit kink the page checks the bend of B2 by eq. 8.1 and raises its mandrel in the schedule where needed (shape code 99); it does not count a cross bar inside the bend (section 5). The bending schedule uses SANS 282 shapes on every route; on this route it takes the 7φ mandrel of Table 8.1N for bars over 16 mm. The SANS 10400-M check under More inputs is the SA building regulation, not part of the Eurocodes. Other annexes may use other values.

3. Stairs as slabs, and their spans

A waist flight is a slab. Most concrete stairs are a sloping slab, the waist, with the steps cast on it. EN 1992-1-1 has no stair clause, so the flight is designed as a slab: a member whose smaller panel dimension is at least five times its thickness (5.3.1(4), 9.3(1)). It spans one way, because its two long edges are free (5.3.1(5)). The steps are load, not strength: the section is the waist h, measured square to the soffit. The calculator designs waist flights; a stepped (sawtooth) soffit with no waist is not modelled.

The effective span. A flight with landings in line spans from wall to wall as one slab. 5.3.2.2(1) gives l_eff = l_n + a₁ + a₂: the clear span plus, at each end, the smaller of half the slab depth and half the support width (Figure 5.4). The page keeps the stair rule of SANS 10100-1 4.9.1.3: the centre of a wall bearing, or half the width of a monolithic beam or wall, at most 900 mm. That is never shorter than 5.3.2.2, so it is on the safe side. Where the wall is as wide as the slab is deep, the two give the same span.

Dog-leg stairs on landings. The flights land on landings (shaded) that span across the stairwell between the walls. Each flight spans onto its two landings; each landing carries its own load and the reactions of both flights. Read literally, 5.3.2.2 would end the flight span h/2 into the landing. But a landing that spans across is not a stiff support: it bends under the flights. The page keeps the rule of SANS 10100-1 4.9.1.3.1: the flight spans to the middle of each landing, at most 900 mm into it. That gives a longer span and a larger moment, on the safe side. Each landing is then designed as a one-way slab across, with its own effective span as an input.

Flights that span across. A flight can instead span across its width: cantilevered from a wall along one side, or between two stringer beams. On its Eurocode 2 route the page reads NOT CHECKED for both; its SANS 10100-1 route designs them (see the SANS twin of this lesson). By hand they are one-way slabs across, per metre of slope, with the same rules: the waist square to the soffit, the steel of 9.3.1.1, and for a cantilever the top bars anchored in the wall and the wall designed for the moment.

4. Actions, bending, shear and span/d

The self-weight on plan. The stair is analysed on its horizontal span with loads per square metre of plan. The sloping waist covers more area than its plan, so its weight per square metre of plan is the thickness divided by cosθ. The steps are triangles of concrete, on average half a riser thick. Reinforced concrete weighs 25 kN/m³ (EN 1991-1-1 Table A.1: 24 for normal concrete plus 1 for the reinforcement). Finishes and the imposed load are given on plan.

On the slope or on plan. A load w per metre of plan on a span L on plan gives the moment w L²/8. Its component square to the slope, w cos²θ per metre of slope, on the sloping length L/cosθ, gives the same moment. So work on plan, and design the section square to the soffit. The shear square to the slope is V cosθ, a little less than the plan shear; the page checks the plan value, on the safe side. The force along an inclined flight is small and is left out, as is usual.

Imposed loads on stairs. EN 1991-1-1 Table 6.1 sorts floors into categories by use, and Table 6.2 gives the loads as ranges; the National Annex picks the value. Stairs appear by name only in category A (homes), at 2.0 to 4.0 kPa. In offices and public buildings the stair is part of the access route: category C3 covers access areas in public and administration buildings, unless your National Annex gives a value for stairs. A stair should never carry less than the floors it serves. ψ₂ of EN 1990 Table A1.1 sets the quasi-permanent load. The page uses the q_k and ψ₂ you give, with EN 1990 6.10: 1.35 G + 1.5 Q.

Bending of the waist. The waist is a slab section 1 000 mm wide and h deep, square to the soffit, with the main bars at the bottom. 6.1 with the rectangular stress block of 3.1.7(3) gives M_Rd. The page keeps x/d at or below 0.448, the limit of 5.5(4) with no redistribution, and designs no compression steel. It uses α_cc = 1.0; with 0.85 the lever arm is a little shorter. In a slab with little steel the difference is a per cent or two.

Shear. A stair waist carries its shear without links: 6.2.2(1) gives v_Rd,c from the depth factor k and the steel ratio ρ_l of the bars that run on past the section, never less than v_min. The page takes V at the support line and at the ends of the flight, without the β that 6.2.2(6) allows for loads near a support and without the cosθ of the slope: on the safe side. In stairs shear rarely governs; a thin waist with links is not practical (9.3.2(1)).

Deflection by span/d. EN 1992-1-1 has no stair rule for deflection either: 7.4.2 applies as to any slab, with the effective span on plan. Eq. 7.16 gives the limit from the required steel ratio at mid-span and K for the structural system (Table 7.4N: 1.0 simply supported, 1.3 for an end span of a continuous slab). SANS 10100-1 allows 15 % more where the flight takes up at least 60 % of the span; EN 1992-1-1 does not, and neither does the page. The page takes K = 1.0 for every end type, on the safe side. The limit rises with the steel provided (eq. 7.17), so more bars can cure a span/d failure; the page caps that factor at 1.5.

5. Steel, kinks, cover and fire

Main and distribution steel. Section across the flight: the main bars run along the stair (dots), the distribution bars across it, on top of them. The main steel lies between A_s,min and 4 % of the section (9.3.1.1(1), which points to 9.2.1.1). A one-way slab needs secondary bars of at least 20 % of the main steel (9.3.1.1(2)): when you add main bars, check the distribution bars again. 9.3.1.1(3) limits the spacing; the page uses the tighter values for areas of maximum moment everywhere, on the safe side.

The kinks in the soffit. Where the flight meets a landing the soffit kinks. At the foot of the flight the soffit corner is convex: a tension bar that follows it is pulled into the concrete, so one bar (B2) may bend round it. At the top of the flight the soffit corner is re-entrant: a tension bar that followed it would try to straighten and burst out through the cover. So the flight bars (B1) run straight on past the corner to the top of the landing, the landing bars (B3) run straight on into the flight, and each is anchored l_bd beyond the corner (8.4). No tension bar runs along the inside face of a re-entrant corner. The page lays the bars out this way and laps B1 with B2 above the lower kink (8.7.3, α₆ 1.5).

Bending a bar at a kink (8.3). A bar is bent round a mandrel at least as large as Table 8.1N (4φ for bars up to 16 mm, 7φ above), so that the bar is not damaged. Where the bar carries its force round the bend, the concrete inside the bend is pressed hard as well. 8.3(3) lets you skip that check only if the mandrel meets Table 8.1N and the bar needs no more than 5φ of anchorage past the bend. A cross bar of at least the bar’s size inside the bend also lets you skip it, but only for a bar away from the side faces. Otherwise eq. 8.1 sets the mandrel. Its a_b is half the bar spacing, or the cover plus φ/2 for a bar next to a side face, so the edge bars need the largest mandrel. A bar bent at a soffit kink carries the full span force on past the bend, so eq. 8.1 applies. In a slab without links, take F_bt from the moment a_l = d further towards mid-span (the shifted envelope of 9.2.1.3(2), with a_l = d by 9.3.1.1(4)). The page checks the bend of B2 at the lower kink by eq. 8.1 (with z = 0.9d and no cross bar counted) and raises the mandrel in the schedule where it must (shape code 99). For a Y20 or larger main bar it schedules at least the 7φ of Table 8.1N.

Top steel at the ends. A flight built into a beam, a wall or a landing across is partly fixed there, whether the analysis says so or not. 9.3.1.2(2) asks for top steel for at least 25 % of the span moment where such partial fixity is left out of the analysis, extending 0.2 times the span from the face of the support; at an end support the moment may be cut to 15 %. The page uses 15 % at a beam or wall and 25 % at a landing across. Where the top face turns at the foot of the flight, the corner is re-entrant in hogging, so the top bars cross again: the landing bars (T1) run on into the flight to its far layer, the flight bars (T1b) on into the landing. The page draws them so but does not check their anchorage beyond the corner. At a simple support the page adds no top steel: where walls above clamp the stair, choose a monolithic end.

Cover and fire. 4.4.1 sets the nominal cover: the larger of the bar diameter (bond, Table 4.2) and c_min,dur for the exposure class (Table 4.4N, class S4 for 50 years), at least 10 mm, plus Δc_dev = 10 mm. Indoors (XC1) the bar size often decides. A stair is an escape route, so it usually needs a fire rating: EN 1992-1-2 5.7.2 and Table 5.8 give, for a simply supported one-way slab, the least thickness h_s and the least axis distance a from the soffit to the centre of the bottom bars. The page checks both with the thinnest part of the stair and does not count the finishes (5.7.1(2) would let them help the separating function).

Which code, and what is left to you. Every check row on the Eurocode 2 route names its EN 1992-1-1 (or EN 1992-1-2) clause. Because EN 1992-1-1 has no stair clause, the geometry of the model comes from SANS 10100-1 4.9: the effective span with landings, the loads on plan and the waist square to the soffit. Some choices are the page’s own: K = 1.0 for every end type, 25 % top steel at a landing across, the maximum-moment spacing everywhere and α_cc 1.0. The bend at the kinks (eq. 8.1), the anchorage of the crossing top bars, flights that span across, the point load Q_k and the walls and beams that carry the stair are yours. The bending schedule uses SANS 282 shapes and bends on every route.