Plastic Moment Mp = Fy Z

M_{p} = F_{y} Z_{x}, ϕ M_{p} = 0.90 F_{y} Z_{x} (compact, adequately braced)

Z is the plastic modulus (sum of first moments of half-areas about the plastic NA). Shape factor Z/S ≈ 1.12–1.15 for W shapes, 1.5 for a rectangle. Compact I-beams can reach Mp.

$Z$: Plastic section modulus
$S$: Elastic section modulus I/c

Step-by-step solved example

W-shape Zx=100 in³, Fy=50 ksi. φMp?

Plastic NA splits the area in two; Mp = Fy Z.

1. Mp
50×100=5000 kip·in=417 kip·ft.
2. φMp
0.90×417=375 kip·ft.

Answer: φMp ≈ 375 kip·ft

10 practice questions

0/10 correct

1.Rectangle b×d: Z=

2.Shape factor of a rectangle is

3.Elastic My=

4.If LTB is not braced, you cannot count on

5.A plastic hinge can form and rotate while

6.S=80 in³, Z=90, Fy=50: My and Mp

7.Compactness (b/t, h/tw) ensures

8.φ=0.90 for flexure of compact steel is

9.Residual moment after unloading a plastified beam is

10.Mechanism for a fixed–fixed beam with mid P: two