Cross-sections

The family

The size

StandardFrom dimensionsIdealDifference

The check

This is where the section meets the internal forces. Enter the axial force and the bending moment from the solver: σ = N/A + M/W, compared with the design resistance.

N in kN (positive = tension) · M in kN·m · γM0 = 1.05

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Which is the lightest section that works?

The question turned around: given the moment and the material above, the lightest section in each family that passes the check. It is the question you actually ask when designing, and comparing the weights is what decides the cost.

Why the real section is not the one in the textbook

The textbook draws sections with sharp corners: rectangles placed side by side. The rolling mill doesn’t — hot steel passing between the rolls does not make right angles, and the standard fixes the radii of those curves together with the dimensions. Those few millimetres shift the second moment of area by a few per cent, and the direction depends on where the curve sits.

The curve adds

I-sections: the root radius fills the corner

Where the web meets the flange there is a root radius r. It fills the re-entrant corner: it is material the textbook drawing leaves out, and it lies far from the neutral axis, so it adds more to the second moment of area than to the area.

IPE 160 · r = 9 mm
4 fillets × 17.4 mm² = 69.5 mm² extra
Ix ideal 834.4 → standard 869.3 cm⁴ (+4.2%)

Across all 66 IPE, HEA and HEB sections the sharp-corner value is below the standard one by between 2.4% and 5.7%: calculating as in the textbook is on the safe side, but it oversizes.

The curve removes

Hollow sections: the rounded corner takes material away

A cold-formed hollow section is bent, not welded at the corners: the outer corners are rounded with a radius of 2t — or 2.5t above 6 mm. Here the curve is on the convex side and removes material exactly where it is furthest from the axis.

SHS 100×100×4 · re = 8, ri = 4 mm
I ideal 236.3 → standard 226.3 cm⁴ (−4.2%)

For hollow sections the sharp-corner value is above the real one by between 2.1% (100×100×2, thin) and 13.7% (20×20×2, small and thick): the thicker the wall relative to the side, the more the corners count. Here calculating as in the textbook is on the unsafe side: you think the section is stiffer than it really is.

And circular tubes? There is no curve to add or remove: the annulus is already the true shape, and the textbook formula matches the standard exactly. It is the control case — if there were a difference there, it would mean the calculation is wrong.

Data checks

Recalculating from the dimensions finds the printed errors

Every value on this page is recalculated from the geometry and compared with the published tables. For the I-sections the major-axis second moment of area agrees within 0.07% on all 66 sections. Where it does not agree, the problem is in the table:

  1. Manuale di Meccanica, Table H.6, octagon. It prints (2 + 2√2)/6 · R4; the correct value is (1 + 2√2)/6 · R4. The difference is 26%.
  2. HEA 200, minor-axis second moment of area. An Italian section handbook gives 1326 cm⁴; the geometry gives 1335.5, a second source 1335, and the section modulus in the same table (133.6 cm³) is consistent with 1336. It is a misprint in the third digit.
  3. Square hollow sections. The same handbook calculates them with sharp corners, ignoring the EN 10219 radii: it overestimates the second moment of area by 2.1% and up to 13.7% on the small sections.
  4. Circular tubes. The same handbook has correct areas but second moments of area that differ by up to 3.7% from the exact formula, which for a circular tube involves no approximation.

What this calculation is, and what it is not

It is an elastic preliminary sizing: maximum stress against fyk/γM0. It tells you in thirty seconds whether a section is in the right size range.

It is not a code check. Above all, instability is missing: a long IPE beam without lateral restraints buckles sideways (lateral-torsional buckling) well before it reaches yield, and a column in compression buckles. Load combinations, deflections, section class, connections and seismic design are also missing.

For a real structure the calculations are signed by a licensed engineer, and this tool is one of the things they use, not a substitute for them.