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.
- Set the checks. Enter the L/n deflection target and allowable bending stress for your material and safety factor.
- Read the results. Read the maximum deflection, allowable deflection, deflection utilization, bending stress, stress utilization and second moment of area.
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.
The result panel now turns those raw numbers into two screening checks. Deflection
utilization compares the elastic deflection with your entered L/n target
such as L/360, L/240 or a stiffer shop limit. Cantilever
checks use the common 2L/n serviceability convention. Stress utilization
compares σ with your entered allowable stress, so you can use yield divided
by your chosen safety factor rather than a hidden material assumption.
If the tube's own weight matters, pair this page with the square tube weight formula, square tube weight calculator, or the broader metal weight calculator, then add that self-weight as a separate distributed load in your beam check. For searchers starting from a table or equation, use the square tube deflection chart or square tube deflection formula pages.
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.
Against an L/360 target, allowable deflection is
1000/360 = 2.78 mm, so deflection utilization is about
36%.
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. With an allowable stress of
150 MPa, stress utilization is about 40%. 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). The calculator also compares δ with your L/n deflection target.
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. Enter an allowable stress after your material safety factor and the calculator reports stress utilization.
What does L/360 mean for square tube deflection?
L/360 means allowable deflection equals the span divided by 360. A 1,000 mm simply-supported span has an L/360 allowance of 2.78 mm. For cantilevers, serviceability tables often use the 2L convention, so the same 1,000 mm cantilever checked at L/360 uses 2,000/360 = 5.56 mm.
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.
- Deflection utilization is a serviceability screen against the entered L/n target. Cantilever checks use the common 2L convention; verify the governing project standard before release.
- Stress utilization uses the entered allowable stress. Pick that value from material yield, safety factor, weld condition and governing design method.
- 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.