beam_model.py. Section properties are nominal and material constants are representative, so treat the answer as a sizing check rather than a design calculation.
A beam's stiffness comes from its second moment of area, I, and for a rectangle that is bd³/12. The depth is cubed and the width is not, so the same cross-section of steel is worth vastly more stood on edge than laid flat. Drag the shape below: the amount of material is held constant to the square millimetre, and the stiffness changes by a factor of twenty.
Why the cube, and not the square. Bending stress grows in a straight line from the neutral axis: a fibre twice as far out is stretched twice as hard. But it also acts on twice the lever arm, so its contribution to resisting the moment goes as y². Add up y² over the whole section — that integral is I — and the outermost material dominates everything. The half of a rectangle nearest the neutral axis contributes only one eighth of the total.
That single fact explains the shapes. An I-beam puts almost all its steel in two flanges as far apart as it dares and leaves a thin web to hold them there. A hole drilled through the web near mid-span removes material that was barely working and is often fine; a notch cut in the flange removes the material doing nearly all of it and is not. And a joist is installed on edge for the same reason a sheet of paper holds nothing flat and a surprising amount folded.
Everything on this page is Euler–Bernoulli beam theory, which assumes the beam is long and slender and that plane sections stay plane. Deflection from bending falls off as 1/d³; deflection from shear falls off only as 1/d. Make a beam deep enough and shear stops being a rounding error and becomes the answer.
The working boundary is the span-to-depth ratio. Above roughly 20:1, shear deflection is under 1% and can be ignored outright. Between 10:1 and 20:1 it is a few per cent — real, usually tolerable. Below 10:1 it is tens of per cent and this calculator is quietly optimistic; below 4:1 the thing is not behaving as a beam at all but as a deep beam or a corbel, where the load arches straight to the supports and a strut-and-tie model is the honest tool.
Two more edges worth knowing. Everything here is elastic — it says nothing about what happens once the material yields, which is where a real design check starts rather than finishes. And long slender beams in bending can fail by lateral-torsional buckling, twisting sideways out of the plane of the load at a moment well below their bending capacity, which no deflection formula will warn you about.
A 203 × 133 UB 25 steel joist spanning 4 m, carrying a uniformly distributed 10 kN/m. I = 2,340 cm⁴, E = 210 GPa.
Mid-span deflection for a UDL on simple supports is 5wL⁴/384EI = 5 × 10,000 × 4⁴ / (384 × 210×10⁹ × 2.34×10⁻⁵) = 6.8 mm. The usual serviceability limit for a floor is L/360 = 11.1 mm, so it passes with room to spare.
Now double the span to 8 m and keep everything else: 4⁴ becomes 8⁴, so the deflection is sixteen times larger — 109 mm against a limit that has only doubled to 22 mm. The span-to-depth ratio is 39:1, so shear is still negligible; the beam is simply far too small. That is the fourth power doing what it always does.
How far the beam sags: governed by stiffness, E × I. A serviceability limit: a beam can be safe yet bounce or crack finishes. δ = k·WL³/EI.
Tension on one face, compression on the other. Peak stress = moment ÷ section modulus: σ = M/Z. The strength check, against yield.
Set purely by how far material sits from the neutral axis. That distance is cubed: I = b·h³/12, so a joist on edge is vastly stiffer than laid flat.
Yield ÷ working stress. Above 1 the static check passes; designers want 1.5–2+ for overload, imperfections and the failure modes this tool does not show.
Bending stiffness follows I = b·h³/12: adding height to a section buys far more resistance than the same amount of width. Joists stand on edge for a reason, and beams are tall rather than wide.
Keep deflection under span/250 generally, span/360 where people will notice a bouncy floor: a 4 m joist may sag 16 mm at span/250, or no more than about 11 mm at span/360. Limits vary by code, application and what the beam supports, so treat any single number as a starting point. Stiffness governs long before strength does.
Either can. A beam might be plenty strong (low stress) but still deflect too much to feel solid, or the reverse. Both are checked independently against their own limits.
An estimation and learning tool. Real structural design accounts for load factors and combinations, shear, lateral-torsional buckling, connections and code requirements this tool does not model. The diagrams stack load, shear, bending moment and deflected shape so you can see where the worst values occur along the span.
Either can. A beam might be plenty strong (low stress) but still deflect too much to feel solid, or vice versa. Both are checked independently against their own limits.
Stiffness depends on the moment of inertia, which grows with the cube of depth. Adding height to a section buys far more bending resistance than adding the same amount of width.
Span/360 is a common reference for floor beams under live load, but acceptable limits vary by code, application, and what the beam supports. Treat any single number as a starting point.
No — it's for estimation and learning. Real structural design accounts for load factors, combinations, buckling, connections, and code requirements this tool doesn't model. Verify against proper methods.
For a simply supported beam, mid-span deflection is δ = PL³ / 48EI for a point load P at the centre and δ = 5wL&sup4; / 384EI for a uniform load w per unit length. E is the material’s stiffness and I the cross-section’s second moment of area. Double the span and a point-loaded beam sags eight times as much.
The maximum bending stress is σ = M c / I, where M is the bending moment, c the distance from the neutral axis to the outer surface and I the second moment of area. For a simply supported beam M = PL/4 for a point load at the centre and M = wL²/8 for a uniform load.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.