Section modulus explained: elastic (Wel) vs plastic (Wpl)

Open any steel section table and you find two section moduli side by side: Wel and Wpl. They are not alternatives you pick by taste - they describe two different states of the same section, and the rules decide which one you are allowed to use.

Elastic modulus: first yield

The elastic section modulus Wel relates moment to the peak bending stress, which occurs at the extreme fibre:

$\sigma_{max} = \frac{M}{W_{el}}, \qquad W_{el} = \frac{I}{c}$

where I is the second moment of area and c the distance to the extreme fibre. The elastic moment capacity - the moment that just brings the outer fibre to yield - is Mel = Wel · fy. Beyond that point the section is still standing; it has only started yielding at the surface.

Plastic modulus: full plastification

Keep loading and yield spreads inward until the whole section has yielded - a plastic hinge. The moment at that state uses the plastic section modulus:

$M_{pl} = W_{pl} \cdot f_y$

Wpl is the first moment of area of the two halves about the plastic neutral axis - geometrically always larger than Wel. That gap is real, usable reserve strength.

The shape factor

Their ratio is the shape factor, α = Wpl / Wel - how much extra moment the section carries between first yield and full plastification. It depends only on shape:

• Rectangle: 1.5
• Typical I-section (major axis): ≈ 1.12-1.18
• Solid circle: ≈ 1.70

An I-beam has a low shape factor because most of its material is already in the flanges, near first yield from the start - there is little left in the middle to mobilise.

Which one are you allowed to use?

This is where section classification comes in. A section can only reach Mpl if it does not buckle locally first:

• Class 1 and 2 (compact) - the section can fully plastify, so you use Wpl.
• Class 3 (semi-compact) - it reaches first yield but buckles before full plastification, so you are limited to Wel.
• Class 4 (slender) - it buckles before first yield; you use an effective modulus, smaller still.

So the table gives both because the section's class - set by its flange and web slenderness - decides your entitlement. Find the class limits in the EC3 reference tables and the Wel/Wpl values for any size in the steel catalogue.