Two-way Slab Deflection

Deflection of a two-way simply supported slab

Take a rectangular slab that rests on all four edges and carries a uniformly distributed load. Its largest deflection sits at the centre and follows δ = α·w·L⁴ / (E·h³). Here w is the load per unit area, L the shorter span, E the modulus of elasticity and h the slab thickness. The coefficient α comes from tables (Marcus, Czerny or Bares) and changes with the aspect ratio Ly/Lx and the bending direction. A square slab gives α ≈ 0.00406 when uncracked and ν = 0.2. Run an example with w = 5 kN/m², L = 4 m, E = 25 GPa, h = 0.12 m and you land at δ ≈ 7 mm.

The Brazilian code NBR 6118 draws the serviceability line at L/300 for visual acceptability, and at L/250 for the total deflection once creep is counted. Tighten that to L/350 or L/500 when the slab carries fragile partitions. Just keep your units lined up: w in N/m², E in Pa, L and h in metres. That way δ comes out in metres.

Applications

Think solid reinforced-concrete slabs in homes and offices, prefabricated panels, ribbed slabs modelled as an equivalent solid plate for early sizing, the slabs of stairs and balconies, and SLS checks you run before committing to a detailed FEM model. When Ly/Lx ≤ 2, the two-way formula spares you the overly cautious assumption that the slab bends in just one direction.

FAQ

Why use h³ instead of I? A strip of unit width has a second moment of area of I = h³/12. The constant 12 simply gets folded into the coefficient α, which leaves the formula working with h³ on its own.

Where do I find α? The Czerny, Marcus and Bares tables list α across a range of Ly/Lx ratios and boundary conditions. For concrete (ν = 0.2) and a square slab on four simple supports, α ≈ 0.00406.

Does it account for cracking? No. Once the section cracks, multiply δ by I_g/I_eff using Branson's approach, or fall back on the bi-linear stiffness in NBR 6118 §17.3.2.

What about long-term effects? Take the instantaneous δ and multiply it by (1 + α_f). For slabs with no compression reinforcement, α_f ≈ 1.32 per NBR 6118 §17.3.2.1.2.