Plastic Section Modulus — Rectangle

Elastic section modulus assumes stress varies linearly across the depth, so only the extreme fibre reaches yield while everything inside is loafing. Push further and the yielded zone spreads inward until the whole section is at $f_y$, compression above the equal-area axis and tension below. The plastic modulus is the first moment of those two half-areas about that axis: $Z = 2 \times (bh/2) \times (h/4) = bh^2/4$. A 50 × 150 mm rectangle gives $Z = 281{,}250$ mm³ against an elastic $S = bh^2/6 = 187{,}500$ mm³.

Their ratio, 1.5, is the shape factor, and it is the reserve strength hiding in every rectangle past first yield. It depends only on the shape: 1.5 for a solid rectangle, about 1.7 for a solid round bar, about 1.12 to 1.18 for a rolled wide-flange section, and roughly 1.27 for a round tube. The I-shape scores lowest precisely because it is already efficient — its material is mostly at the extreme fibre, so there is little understressed core left to recruit.

Two things to keep straight. The plastic axis is the equal-area axis, which coincides with the centroid only when the section is symmetric about the bending axis; for a tee or a channel the two are different lines, and using the elastic centroid gives the wrong $Z$. And the reserve is only available to a compact section that can actually reach the fully plastic state without buckling locally or rolling over sideways first. A slender web will fold at a stress well below $f_y$, and then neither $Z$ nor $S$ tells you anything useful.