Flat slabs

How a flat slab on columns is designed with the STATIX flat slab calculator on its Eurocode 2 route: the equivalent frame of EN 1992-1-1 Annex I, the load arrangements, column and middle strips, the moment at edge and corner columns, punching to 6.4 with links, span/d and the detailing rules of 9.4.

1. Learning outcomes

After this lesson you can:

set up the equivalent frames of a flat slab in both directions (Annex I.1.2);
apply the load arrangements of 5.1.3 and take the support moment at the column face;
share the frame moments between column and middle strips and design the steel, with A_s,min and the bar spacing;
check the moment an edge or corner column can take, and place its steel within b_e;
check punching at u₁ = 2d with β, and design punching links by 6.52;
check span/d by 7.4.2 and detail the slab at its columns by 9.4.

2. Before you start

RC Design 1 (Eurocode): bending and steel (lesson 1), two-way slabs (lesson 6) and punching (lesson 7).
The flat slab calculator, set to Eurocode 2 (EN 1992-1-1).
A pocket calculator and about 60 minutes.
The same lesson on the SA code: RC Design 2, lesson 1 (SANS 10100-1).

What the flat slab page’s Eurocode 2 route uses: EN 1990 6.10 with the recommended factors, 1.35 G + 1.5 Q (the page does not offer 6.10a and 6.10b), and 1.35 G on the spans without imposed load; the imposed load on all spans, odd spans, even spans and every two adjacent spans; frames of the full panel width on gross sections, the columns above and below as springs 4EI/H fixed at their far ends, E_cm of Table 3.1; the support moments at the column face; a column strip half the smaller span wide with 75 % of the hogging and 55 % of the sagging; bending with α_cc 0.85 (K′ 0.167, z with 1.134, z ≤ 0.95d); A_s,min of 9.2.1.1 on d; bar spacing at most 2h and 250 mm everywhere; d with 12 mm bars for bending; for punching d from the top bars placed and ρ_l at each column (interior, edge, corner) from the top bars at that column (6.4.4(1)); the edge moment limit of Annex I.1.2(5); punching by 6.4 with the β values of 6.4.3(6), v_Rd,max = 0.4νf_cd with f_cd = f_ck/γ_c (α_cc 1.0 for shear) and links by 6.52 (the link steel at f_yk 450 MPa at most, on the safe side); span/d by 7.4.2 with K 1.2; the cover by 4.4.1 (c_min,dur of Table 4.4N, class S4, c_min,b of a 12 mm bar, Δc_dev 10 mm: bars over 15 mm need more cover, see Example A). Other annexes may use other values.

3. The slab as frames

Why flat slabs. A flat slab sits directly on its columns. It is quick to build and leaves a flat soffit. But the whole floor load gathers into a small area at each column, so punching shear and the moments next to the columns decide the design. RC Design 1 (Eurocode) lesson 7 introduced punching; this lesson designs the whole slab.

The equivalent frame. EN 1992-1-1 Annex I.1.2 divides the slab into frames each way: a line of columns and the slab between the centre lines of the panels on each side (l₂). Each frame is analysed as a continuous beam on its columns, with the total load on the panels, first in one direction, then in the other. For vertical load the stiffness is the full panel width on the gross section. The calculator takes the columns above and below as rotational springs fixed at their far ends; set the storey heights to 0 for knife-edge supports.

Load arrangements. The imposed load may be on some spans and not others. EN 1992-1-1 5.1.3 recommends two arrangements for buildings: alternate spans loaded, and any two adjacent spans loaded, with the design permanent load γ_GG_k on every span. The calculator runs all spans, odd spans, even spans and each pair of adjacent spans, with 1.35 G_k on the unloaded spans, and keeps the worst moment at each section. Two loaded neighbours give the largest hogging at the column between them.

The design hogging moment. The peak moment over the column centre is not used. Where a slab is monolithic with its support, EN 1992-1-1 5.3.2.2(3) takes the design moment at the face of the support, and its note asks that it is not less than 0.65 of the full fixed-end moment. The calculator takes the hogging at both faces of every column and reports the larger. Check the 0.65 note by hand; the page does not.

4. Strips and steel

Column and middle strips. Annex I.1.2(3) divides each panel into a column strip, a quarter of the smaller span each side of the column line, and the middle strips beside it. Table I.1 gives the share of the frame moment for each strip as a range. The calculator uses 75 % of the hogging and 55 % of the sagging in the column strip, inside those ranges. Steel is then designed per metre of each strip.

Minimum steel and spacing. Each strip needs at least A_s,min of 9.2.1.1(1) on its effective depth. The bar spacing is limited by 9.3.1.1(3): 3h and 400 mm for the main bars in general, but 2h and 250 mm in areas of maximum moment or concentrated load. Near flat slab columns that is most of the slab, so the calculator applies 2h and 250 mm everywhere, on the safe side.

Edge and corner columns. A flat slab can pass only a limited moment into a column at its edge, through a narrow width b_e around the column (Figure 9.9). Annex I.1.2(5) limits it to 0.17 b_ed²f_ck unless there is an edge beam designed for torsion, and asks for the end-span sagging to be adjusted. The calculator cuts a larger end moment to the limit and adds 0.6 of the difference to the end span, a practice rule. 9.4.2(1) asks for the steel that carries this moment to sit within b_e.

5. Punching and deflection

The basic control perimeter. EN 1992-1-1 6.4.2 checks the shear stress on a perimeter at 2d from the column face, with rounded corners, d being the mean of the two layers. For an interior rectangular column u₁ = 2(c₁ + c₂) + 4πd. The factor β allows for the moment the column takes. Where the structure is braced and adjacent spans differ by no more than 25 %, 6.4.3(6) allows 1.15 interior, 1.4 edge and 1.5 corner. The calculator always uses these values. It does not check the two conditions, so check them by hand.

Resistance without links. 6.4.4(1) gives v_Rd,c from the depth factor k and the tension steel ratio ρ_l, the geometric mean of the two directions, taken over the column width plus 3d each side. The calculator takes ρ_l from the top bars of the column strips. More top steel over the column raises v_Rd,c, but only by its cube root.

Punching links. Where v_Ed on u₁ is more than v_Rd,c, 6.4.5(1) lets links carry the rest: 75 % of the concrete resistance stays, and each perimeter of links adds its share. The same area goes on every perimeter. At the column face the stress on u₀ must not pass v_Rd,max, whatever the links. A slab with punching links must be at least 200 mm deep (9.3.2(1)).

How far the links go. 6.4.5(4) finds the perimeter u_out,ef where the concrete alone is enough. The last perimeter of links must be no more than 1.5d inside it. 9.4.3(4) puts the first perimeter no more than d/2 from the column face; 9.4.3(1) sets the perimeters at most 0.75d apart, at least two of them, and the legs at most 1.5d apart around a perimeter inside u₁ (2d outside it). Eq. 9.11 (9.4.3(2)) sets the least area of one leg. The calculator gives the area and the number of perimeters; How it is calculated gives the legs per ring.

Deflection by span/d. 7.4.2 gives the limiting span/d from the steel ratio at mid-span by eq. 7.16, with K = 1.2 for a flat slab (Table 7.4N), on the longer span. It is raised when more steel is provided than needed, and cut by 8.5/l where the longer span is over 8.5 m and the slab carries partitions liable to be damaged (7.4.2(2); the page applies it to every span over 8.5 m, on the safe side). At low steel ratios eq. 7.16a gives a large limit; a lightly reinforced flat slab can still deflect more than you like, so lesson 6 calculates deflection directly.

Drops, heads and openings, in outline. A drop or a column head thickens the slab where the shear and hogging are highest; an opening within 6d of a column cuts part of the control perimeter. The calculator takes drops, heads, openings and cantilever edges; this lesson works the plain slab on the Eurocode 2 route and leaves them as options to try.

6. Detailing at the columns

Top steel over internal columns. The hogging is highest close to the column. 9.4.1(2) asks for half of the top steel of the whole frame width, A_t, to be placed within 0.125 of the panel width on each side of the column, unless serviceability is checked in full. 9.4.1(3) asks for at least two bottom bars each way to pass through the column, to hold the slab if punching occurs. The calculator gives the steel per metre of strip; place it by hand (Example A shows how).

Free edges. Along a free edge a slab needs bars along the edge and bars across it, anchored round the edge, usually as U-bars (9.3.1.4, Figure 9.8). The normal slab bars may serve. At edge and corner columns the bars that carry the moment into the column go within b_e (9.4.2(1)).