How to use this calculator
- Pick the load case. Choose simply supported with a central point load, or a cantilever with the load at the free end.
- Enter the section size. Enter the outside width b, outside height h and wall thickness t of the tube.
- Enter the span and load. Enter the span (or cantilever length) L, the point load F, and the material’s Young’s modulus E.
- Read the results. Read the maximum deflection δ, maximum bending stress σ and the second moment of area I.
How it works
A square or rectangular hollow section (HSS tube) used as a beam resists bending
through its second moment of area. For a hollow rectangle it is the
outer rectangle minus the inner void:
I = (b·h³ − bᵢ·hᵢ³) / 12,
with bᵢ = b − 2t and hᵢ = h − 2t. The depth h
enters as a cube, so a deeper tube is dramatically stiffer for the same wall.
The maximum elastic deflection then depends on how the beam is supported. Simply
supported with a central point load:
δ = F·L³ / (48·E·I).
A cantilever with the load at the free end:
δ = F·L³ / (3·E·I).
The peak bending stress is
σ = M·c / I,
with c = h/2 and the maximum moment M = F·L/4 (simply
supported) or M = F·L (cantilever). Because N·mm / mm⁴ × mm = MPa, the
stress comes straight out in MPa.
Worked example
Verified against the live calculator
A 50×50×3 mm steel tube (E = 200 GPa), simply supported over
L = 1,000 mm with a 2,000 N central load. The inner
dimensions are bᵢ = hᵢ = 50 − 2×3 = 44 mm, so
I = (50·50³ − 44·44³)/12 = (6,250,000 − 3,748,096)/12 ≈ 208,492 mm⁴.
The deflection is
δ = 2000 × 1000³ / (48 × 200,000 × 208,492) ≈ 0.999 mm.
The moment is M = F·L/4 = 500,000 N·mm, so the bending stress is
σ = 500,000 × 25 / 208,492 ≈ 59.95 MPa. The calculator returns exactly
these numbers.
Frequently asked questions
How do I calculate the deflection of a square tube?
First find the second moment of area of the hollow section, I = (b·h³ − bᵢ·hᵢ³)/12, where bᵢ = b − 2t and hᵢ = h − 2t. Then for a simply-supported beam with a central point load, δ = F·L³/(48·E·I); for a cantilever with an end load, δ = F·L³/(3·E·I). For a 50×50×3 mm steel tube spanning 1,000 mm under a 2,000 N central load, I ≈ 208,492 mm⁴ and δ ≈ 1.0 mm.
What is the second moment of area of a rectangular tube?
A rectangular hollow section is the outer rectangle minus the inner void, so I = (b·h³ − bᵢ·hᵢ³)/12 about the horizontal axis, with bᵢ = b − 2t and hᵢ = h − 2t. The depth h enters as a cube, so making the tube taller (greater h) stiffens it far more than making it wider for the same wall thickness.
Simply supported or cantilever — which load case do I use?
Use simply supported (central point load) when the tube rests on a support at each end and the load sits at mid-span — δ = F·L³/(48·E·I). Use cantilever when one end is fixed (welded or clamped) and the load hangs off the free end — δ = F·L³/(3·E·I). For the same span and load the cantilever deflects 16× as much, because it is far less restrained.
How do I find the bending stress in the tube?
Bending stress is σ = M·c/I, where M is the maximum bending moment and c = h/2 is the distance to the extreme fibre. The moment is M = F·L/4 for a simply-supported central load, or M = F·L at the fixed end of a cantilever. Compare σ against the material yield strength with a safety factor.
Which way should I orient a rectangular tube?
Orient the larger dimension (greater depth) vertically, in line with the load. Because depth enters the inertia as h³, a 40×80 tube stood on its 40 mm face is far stiffer and stronger than the same tube laid on its 80 mm face — same material, very different deflection. This calculator bends about the axis parallel to the width b.
Does the calculator include the tube's own weight?
No — it models a single point load and ignores self-weight, which keeps the formulas exact and transparent. For long, lightly-loaded spans the beam's own mass matters; add it as a separate distributed-load deflection. The result is also elastic (Euler–Bernoulli) and valid only while the material stays below yield.
Method & assumptions
- Euler–Bernoulli elastic beam theory — small deflections, plane sections remain plane, and the material stays below yield.
- Single concentrated point load only: at mid-span (simply supported) or at the free end (cantilever). Self-weight is ignored — add it as a distributed load for long, lightly-loaded spans.
- Bending is about the axis parallel to the width b; the depth h resists bending. Orient the larger (deeper) dimension vertically to use the section efficiently.
- I = (b·h³ − bᵢ·hᵢ³)/12 assumes a true hollow rectangle with uniform wall t and sharp corners; real HSS has rounded corners, so the catalogue I is slightly different.
- No shear deflection, local buckling, web crippling or stress concentration at the load or supports — slender beams only.