A cantilever is a beam held at one end and free at the other, like a balcony slab or the arm of a road sign, and the textbook gives its tip deflection as PL³/3EI under a load P at the free end. Make it twice as long and that load bends it eight times as far. Its allowance grows with the length too, so it uses four times as much of a limit written as a fraction of length, and eight times under a uniform load. We checked the formula against our own finite-element model of a solid rectangle. At a span of ten depths it was within 1.4% in every case we ran. Near three depths it was out by about a tenth.
Length controls the share of a limit that a cantilever uses
In PL³/3EI, E is the material's elastic modulus and I the section's second moment of area. Under a uniform load w per metre the tip deflection is wL⁴/8EI. The tables quoted below write limits as the length divided by a number, L/n, so the share used is the deflection times n, divided by L. Double the length and the share rises four times under a tip load, and eight times under a uniform load with w per metre unchanged, so the total load doubles as well.
On our arithmetic, with E 200 GPa, I 1,000 cm⁴ and a 5 kN tip load: at 2 m the tip moves 6.67 mm against a Length/180 allowance of 11.11 mm, 60% of it. At 4 m it moves 53.33 mm against 22.22 mm, 240%. The longest beam within L/n, √(3EI/nP), moves only with the square root of n: 2.58 m at L/180 and 1.83 m at L/360.
Depth governs by the cube only in a solid bar
Doubling the depth of a solid rectangle multiplies I by eight and cuts a tip load's deflection to an eighth. It also doubles the weight. A beam's own weight is a uniform load, so its sag goes with weight divided by I and falls only to a quarter.
In an I-section the eight shrinks. We worked a synthetic section, 400 mm deep, 180 mm wide, with 13.5 mm flanges and an 8.6 mm web (not a catalogue size). Doubling its depth with the same flanges and web multiplies I by 4.95 and the weight by 1.43, so tip-load deflection falls to 0.20 of its old value and self-weight deflection to 0.29.
Hong Kong's steel code lists cantilevers apart from its span limits
Clause 5.2 of Hong Kong's Code of Practice for the Structural Use of Steel 2011 (2023 Edition) reads: "For typical structures, the deflection limits in Table 5.1 are recommended." Under "Vertical deflection of beams due to imposed load", the table gives Length/180 for cantilevers, Span/360 for beams carrying plaster or other brittle finish and Span/200 for other beams. The heading names imposed load only. A note under the table reads: "Exceedance of the above limit is not acceptable unless a full justification is provided."
A draft Table 1604.3 filed with the Florida Building Code's 2007 supplement reaches the same figure for two rows by another route. Its footnote i reads: "For cantilever members, l shall be taken as twice the length of the cantilever". Floor members under live load take l/360, which becomes L/180; so do roof members supporting a plaster ceiling. Its other entries, read the same way, run from L/120 to L/60. We could not open the current IBC or Eurocode 3, so we quote neither.
Until 3 November 2026 our cantilever deflection calculator judged a cantilever against span/250 and span/360, neither of them a cantilever line in either table. At 2.5 m the worked beam showed red, though its 13.02 mm met Length/180's 13.89 mm. The fix dated that day compares only with Length/180, shows the length-to-deflection ratio, refuses negative loads and stops comparing past a sag of a tenth of the length. With the inputs above, 2.58 m gives 14.31 mm, within the 14.33 mm allowance at 100%.
Shear claims about a tenth of the sag in a short, deep rectangle
PL³/3EI counts bending only, and shear deflection grows as a beam gets shorter for its depth. We modelled a solid rectangle in plane stress, clamped at the root. For the I-section above we used a shear term alone, with the web carrying the shear and Hong Kong's steel moduli.
| Span ÷ depth | Rectangle, tip load (%) | Rectangle, uniform load (%) | Synthetic I-section, tip load (%) |
|---|---|---|---|
| 3 | 7.9 | 11.6 | 36.9 |
| 5 | 2.7 | 4.0 | 13.3 |
| 10 | 0.6 | 0.9 | 3.3 |
| 15 | 0.2 | 0.3 | 1.5 |
Each figure is extra deflection beyond the formula; the rectangle's figures held within about a tenth of a point on a finer mesh. A cut-off of ten depths for ignoring shear is cautious for a solid rectangle. On our I-section, shear added 10% near 5.8 depths under a tip load, about four times what the same shear term gives a rectangle of equal span and depth.
Wall conditions steer a stubby beam as much as shear does
Beam theory assumes a rigid root. A second model, its root held flat but free to strain through the depth, gives more. At three depths the tip-load error rises from about 8% to about 10%, and the uniform-load error from about 12% to about 17%. By ten depths the two differ by under half a point. Below three depths that model depended on the mesh, so we give no figure. A real wall can rotate and settle, like neither model.
Yield pre-empts large deflection in a solid steel bar
The formula also assumes a small deflection. We solved the exact large-deflection curve for a tip load and checked it against SOFiSTiK's Benchmark No. 7, a 10 m cantilever whose tip deflection we matched within 0.1% of its quad-element result. PL³/3EI over-reads by 0.11% when it predicts a sag of L/30, 1% near L/10 and 10% at L/3.1.
A solid steel bar yields well before then. At first yield under a tip load, a rectangle's tip sag as a fraction of its length is two thirds of yield stress over E, times span over depth. With 355 MPa and 205 GPa that is L/87 at ten depths and L/29 at thirty, on our arithmetic.
Cracking, creep and the chosen modulus stay outside the formula's control
Deflection scales with 1/E. Section 3 of the Hong Kong steel code gives E as 205,000 N/mm², which gives 2.4% less deflection than 200 GPa. An April 2026 amendment to the same code, citing Hong Kong's concrete code, tabulates a short-term modulus of 22.2 kN/mm² for C30 and 30.0 for C60, so taking 30 for a C30 member understates deflection by a factor of 1.35.
Those figures are short-term and uncracked. A cracked member loses stiffness, and a loaded one creeps over the years. Neither is in PL³/3EI, and no source we opened gives a figure for either.
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