Square tube deflection, explained

Square and rectangular tube (HSS) is the default DIY-and-fab beam — racks, gantries, trailers, workbenches, machine frames — and nearly every design question about it reduces to one number: the second moment of area of the hollow section,

I = (b·h³ − bᵢ·hᵢ³) / 12,  bᵢ = b − 2t, hᵢ = h − 2t

Outer rectangle minus inner rectangle. Feed it the load-case formula and the two service questions answer themselves:

δ = F·L³ / (48·E·I) (center point)   δ = 5·w·L⁴ / (384·E·I) (uniform)

Worked example — 50×50×3 over a metre

A 50×50×3 mm steel tube (E = 200 GPa), simply supported over L = 1,000 mm, with F = 2,000 N (≈204 kg) at center:

I = (50·50³ − 44·44³)/12 = 208,492 mm⁴

δ = 2,000 × 1,000³ / (48 × 200,000 × 208,492) = 1.0 mm

Bending stress at the same point is σ = M·c/I = 60 MPa — a quarter of mild-steel yield, while the deflection sits at a stiff L/1000. The square tube deflection calculator runs both (plus cantilever and uniform-load cases), and the deflection chart tabulates common sizes.

Bigger beats thicker

Depth enters the stiffness as h³; wall thickness only linearly-ish. From the same 50×50×3 baseline:

• Double the wall (50×50×6): I → 347,072 mm⁴ — 1.66× stiffer for 87% more steel.
• One size up (60×60×3): I → 371,412 mm⁴ — 1.78× stiffer for 21% more steel.

Same lesson as every beam: when deflection governs, buy section depth, not wall. Thick wall earns its keep elsewhere — bearing at connections, weld capacity, dent resistance and buckling of very thin walls.

The span cube (and fourth power)

Deflection grows with L³ for point loads and L⁴ for distributed ones. Stretch the worked example's span by half (1.0 m → 1.5 m) and the same load deflects 3.375× as much: 3.4 mm against an L/360 allowance of 4.2 mm. Still inside — but the margin collapsed from 2.8× to 1.2×, and the L⁴ scaling of distributed loads makes long spans worse still. This is why "it held fine on the short bench" fails on the long one.

Common mistakes

• Using the solid-rectangle I. b·h³/12 without subtracting the inner rectangle overstates a 50×50×3 tube's stiffness 2.5× — the single most common error in forum threads.
• Same-weight solid bar "for strength". The tube is 7.9× stiffer than the equal-weight solid square — tube is not the compromise, it is the optimum.
• Forgetting the load case. A cantilever deflects 16× more than the same simple span and load — fix the support assumption before arguing about size.
• Checking strength only. The example carries stress with 4× margin while sitting near serviceability limits at longer spans — read deflection against the L/360-family limits, not just yield.

For rectangular sections oriented the strong way, the same calculator handles b ≠ h; the section modulus page covers the strength property, and the beam load capacity calculator runs the question in reverse — the load a chosen tube can carry at a target deflection.

Frequently asked questions

How do I calculate square tube deflection?

Pick the load case formula — δ = F·L³/(48·E·I) for a center point load on a simple span, 5·w·L⁴/(384·E·I) for uniform load — with the tube's second moment of area I = (b·h³ − bᵢ·hᵢ³)/12, where the inner dimensions are the outer ones minus twice the wall. A 50×50×3 tube spanning 1 m under 2 kN at center deflects 1.0 mm.

How much stronger is a bigger tube vs a thicker wall?

Size wins decisively. On a 50×50×3 tube, doubling the wall to 6 mm costs ~87% more steel for 1.66× the stiffness; going to 60×60×3 costs ~21% more steel for 1.78× the stiffness. Depth enters as h³ — buy section size, not wall, when deflection governs.

Why use tube instead of solid bar?

Bending stresses live at the extreme fibres; the middle barely works. A 50×50×3 tube has 7.9× the bending stiffness of a solid square bar of the same weight. That is the entire argument for hollow structural sections.

How much deflection is acceptable?

Serviceability limits are span ratios: L/360 under live load is the customary floor rule, L/240 total, L/180 for rougher service. The 1 mm worked-example deflection on a 1,000 mm span is L/1000 — stiff. See the deflection limits guide for the full set.