Two-Way & Flat Slabs Notes

Last reviewed 16 Sept 2026 · 6 min read

Two-way action

A slab supported on all four sides with $l_{y} / l_{x} \leq 2$ bends in both directions. Load is shared between the two spans; the short span carries more because it is stiffer (deflection varies with span⁴).

Rankine–Grashoff theory

Treat two crossing strips at the centre with equal deflection. For a uniformly loaded slab simply supported on four edges:

w_{x} = w \frac{l _{y}^{4}}{l _{x}^{4} + l _{y}^{4}}, w_{y} = w \frac{l _{x}^{4}}{l _{x}^{4} + l _{y}^{4}}

M_{x} = \frac{w _{x} l _{x}^{2}}{8}, M_{y} = \frac{w _{y} l _{y}^{2}}{8}

The theory ignores the torsional stiffness of the slab, so it overestimates moments. Marcus's correction reduces them. IS 456 Annex D gives coefficients based on yield-line and elastic analyses, which are used in practice.

Simply supported slabs — corners not held down

When corners are free to lift (no torsional restraint, no corner steel), IS 456 Table 27 gives:

M_{x} = α_{x} w l_{x}^{2}, M_{y} = α_{y} w l_{x}^{2}

(both multiplied by $l_{x}^{2}$ , the short span).

Code ProvisionIS 456 Table 27 — simply supported slabs, corners not held down

$l_{y} / l_{x}$	1.0	1.1	1.2	1.3	1.4	1.5	1.75	2.0
$α_{x}$	0.062	0.074	0.084	0.093	0.099	0.104	0.113	0.118
$α_{y}$	0.062	0.061	0.059	0.055	0.051	0.046	0.037	0.029

At least 50% of the tension steel should extend to the supports; the rest may be curtailed within $0.1 l$ of the supports.

Restrained slabs — corners held down

For slabs cast monolithically with beams, corners are prevented from lifting and edges may be continuous or discontinuous. IS 456 Table 26 lists nine cases (interior panel, one short edge discontinuous, one long edge discontinuous, two adjacent edges discontinuous, …, four edges discontinuous) with negative coefficients (at continuous edges) and positive coefficients (at mid-span) for both directions:

M = α w l_{x}^{2}

Code ProvisionIS 456 Table 26 — two cases, selected values

Case 1: Interior panel (all edges continuous)

$l_{y} / l_{x}$	1.0	1.1	1.2	1.3	1.4	1.5	1.75	2.0	Long span (all ratios)
Short span, negative	0.032	0.037	0.043	0.047	0.051	0.053	0.060	0.065	0.032
Short span, positive	0.024	0.028	0.032	0.036	0.039	0.041	0.045	0.049	0.024

Case 9: Four edges discontinuous (corners held down)

$l_{y} / l_{x}$	1.0	1.1	1.2	1.3	1.4	1.5	1.75	2.0	Long span (all ratios)
Short span, positive	0.056	0.064	0.072	0.079	0.085	0.089	0.100	0.107	0.056

The long-span coefficient does not change with $l_{y} / l_{x}$ — it depends only on the edge conditions.

Middle strips, edge strips and corner steel

Each span is divided into a middle strip (width three-quarters of the span) and two edge strips (each one-eighth of the span).
Reinforcement from the coefficients goes in the middle strips; edge strips get only minimum steel (0.12% for HYSD).
Torsion reinforcement at corners (where corners are held down):
- At a corner where both edges are discontinuous: provide top and bottom mesh, each layer in both directions, of area three-quarters of the area required for the maximum mid-span moment in the slab, over a distance $l_{x} /5$ from the edges.
- At a corner where one edge is discontinuous: half of the above.
- At a corner where both edges are continuous: none.

Loads on supporting beams

The slab load is transferred to beams by drawing 45° lines from the corners: the short beams receive triangular loads and the long beams trapezoidal loads. Equivalent UDLs for bending moment: triangular load $w l_{x} /3$ ; trapezoidal load $\frac{w l _{x}}{2} [1 - \frac{1}{3} (\frac{l _{x}}{l _{y}})^{2}]$ (per unit length of beam, with $w$ in kN/m²).

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Two-Way & Flat Slabs