Two-way slabs

Most floors are slabs carried on beams or walls on all four sides. This lesson shows how such a slab shares its load between two directions, and how to find its moments, steel, corner bars, shear and deflection to SANS 10100-1.

1. Learning outcomes

After this lesson you can:

tell one-way from two-way slabs and explain why the short span works harder;
find the design moments from moment coefficients for the edge conditions;
design the two layers of bottom steel, with minimum steel and spacing;
decide when corners need torsion steel and how much;
check shear and span/depth, and compare with the STATIX plate analysis.

2. Before you start

Lesson 1 (K, z, A_s) and lesson 4 (span/depth).
The free STATIX slab calculator, open in a second tab.
A pocket calculator and about 60 minutes.

3. One-way or two-way?

A slab supported on all four sides carries its load to the nearest supports. A long, narrow panel sends nearly all of it the short way, like a wide beam (one-way). A panel closer to square shares it between both directions (two-way). A common dividing line is l_y/l_x = 2.

Why the short span works harder. Picture two strips crossing at the centre; both must sag the same amount there. A short strip is much stiffer than a long one, so it takes the bigger share of the load. That is why two-way slabs get more steel across the short span.

4. Edges and corners

Edge conditions, as you set them in STATIX (ss, fix, free). A continuous edge runs on over a beam or wall into the next panel, so it has a hogging moment and needs top steel. A simply supported edge only props the slab. A free edge has no support at all.

What happens at the corners. A slab dishing down on four simple supports tries to curl its corners up. If they are free to lift, the moments follow SANS 10100-1 Table 14. If walls or beams hold them down, the corners twist and crack, and need torsion steel (4.4.4.2).

5. Moments from coefficients (SANS 10100-1, 4.4.4)

For rectangular panels with a uniform load, the code gives the mid-span moments per metre width as a coefficient times n l_x², where n is the design load per square metre and l_x the short span:

M_sx = α_sx n l_x² (short span) M_sy = α_sy n l_x² (long span)

Table 14 (4.4.4.1): simply supported on four sides, corners free to lift.
Table 15 (4.4.4.2): restrained slabs, corners held down, with nine cases by which edges are continuous. These give mid-span (sagging) and support (hogging) coefficients β.

Moment coefficients for a slab simply supported on four sides with free corners: M = α n l_x² per metre width. The curves come from α_sx = r⁴/(8(1 + r⁴)) and α_sy = r²/(8(1 + r⁴)) with r = l_y/l_x, which give the values of SANS 10100-1 Table 14. As the panel gets longer, the short span takes almost all the moment.

6. The steel

Two layers of bottom steel. The bars that span the short way do the most work, so they go in the outer layer where the effective depth is largest. The long-span bars sit on top of them, one bar diameter higher.

Each layer is designed per metre width exactly like the beam of lesson 1 (b = 1 000 mm). Then:

minimum steel 0.13% of b h (f_y ≥ 450 MPa), in both directions;
bar spacing not more than 3h or 400 mm, whichever is less;
for crack control, a spacing limit of 3d or 750 mm, which you need not check when h ≤ 200 mm or the steel is under 0.3%.

Middle and edge strips (SANS 10100-1, 4.4.4.2). The coefficient moments apply to the middle three quarters of the panel; the edge strips, one eighth each side, need only minimum steel (and the torsion steel at the corners). Many designers simply run the middle-strip steel right across.

Corner torsion steel (SANS 10100-1, 4.4.4.2.2). Where corners are held down, each corner gets a mesh in the top and in the bottom over a square l_x/5 wide. The amount depends on how many of the two edges meeting there are continuous.

7. Shear and deflection

Shear in slabs (SANS 10100-1, 4.4.5). The load flows out to the edges. SANS gives no shear coefficients for two-way slabs, so STATIX takes half the load on a strip across the short span, which is on the safe side. Slabs are designed so that v stays below v_c and no links are needed.

Span/depth works as in lesson 4, on the short span. For slabs STATIX takes the service stress as f_s = (2/3) f_y A_s,req/A_s,prov, and the basic ratio from Table 10: 16 for a truly simply supported panel, 20 when nominally restrained, higher when edges are continuous.

8. What STATIX adds, and flat slabs

What STATIX does. By default the slab calculator solves the panel as a plate (a fine grid) and designs from the moment field, with the Wood-Armer method to combine bending and twisting. For a panel simply supported on four sides with free corners it uses the larger of Table 14 and the plate result. You can switch it to the SANS coefficients (Table 15) to compare with a hand calc.

Flat slabs sit straight on columns, with no beams. They are designed in column strips and middle strips, and the slab around each column must be checked for punching shear (lesson 7). STATIX has a separate Flat Slab calculator for this: equivalent frames, strips, punching at interior, edge and corner columns, span/depth and a schedule.