Beam dimensionsDepth in a solid slab, width in a ribbed slab

Why a 'governing segment'? (30 seconds)

• A continuous beam crosses several spans — and each span has its own length and load.
• Deflection grows with length and with load → compare L₀ · ∛f_ser of every segment, and the largest governs.
• Not always the longest span: a short span with a heavy load can win.

Solid slab: a beam 30 wide — what is the depth?

A beam on 3 supports (3.4 + 4.6 m), width b = 30. A slab rests on the 4.6 segment: q = 2.3 t/m; on the 3.4 segment — none.

• L₀
  0.8 · 340 = 272 · 0.8 · 460 = 368 cm
• h₀
  368 / 10 = 36.8 → round up in steps of 5, minimum 30 → h₀ = 40 cm
• S.w
  2.5 · (40/100) · (30/100) = 0.3 t/m
• f_ser
  0 + 0.3 = 0.3 · 2.3 + 0.3 = 2.6 t/m
• Governing segment
  272 · ∛0.3 = 182.09 · 368 · ∛2.6 = 506.03 → the 4.6 segment
• K₁₂
  5.25 · ∛(30 / 2.6) = 11.86
• Required h
  368 / (1 · 11.86 · 1) = 31.03 → 32 ≤ 40 passes ✓

That's it. Beam 30/40.

And if the exercise gives h = 30 ('check')? S.w = 0.225 → f_ser = 2.525 → K₁₂ = 11.98 → h_req = 30.72 → 31 > 30 — does not pass. In a 'check' question this is the answer; in a 'design' question — increase in steps of 5 cm above the required h and calculate again.

Units: b and h in cm, f_ser of a beam in t/m.

Ribbed slab: hidden beam — looking for the width

In a ribbed slab the beam is 'swallowed' by the slab thickness: h = 33 cm is given. In the governing segment q = 6.403 t/m, L₀ = 413 cm. In a hidden beam f_ser is only the q from the load transfer.

• Beam depth = slab thicknessh = 33 cm
• Load coefficientK₁₂ = L₀K₁₁ · h · K₁₃ = 4131 · 33 · 1 = 12.52
• Required widthb_req = f_ser · (K₁₂5.25)³ = 6.403 · (12.525.25)³ = 86.84 cm
• Beam width (steps of 5 cm)b_req = 86.84 → b = 90 cm ✓

That's it. A hidden beam 90 cm wide — wide, but it does not protrude from the slab.

4 pitfalls

• Forgetting the beam's self-weight. In a solid slab f_ser of the beam = q + S.w; only in a hidden beam — q alone.
• The governing segment = the longest. Compare products, not lengths.
• Hidden-beam width in whole cm. Round up to steps of 5 cm.
• Calculating depth in a ribbed slab. The depth is already given — the slab thickness.