CIV-22

Plastic moment Mp

Mp = fy Z. For a rectangle Z = bh²/4 so Mp = fy b h²/4.

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PlasticityPlastic modulus Z

Governing equation

M_p=f_y Z=f_y\,bh^2/4

$h \in \left[80, 900\right]$

where

f_y: Yield stress (MPa)
b: Width (mm)
h: Depth (mm)
M_p: Plastic moment (kN·m)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

M_p=f_y Z=f_y\,bh^2/4

$h \in \left[80, 900\right]$

$h = 80$ $M_p = 113.6$

Along the curve

②

Domain of the variable — blue

h_{\min} = 80,\quad h_{\max} = 900

$h \in \left[80, 900\right]$

$h = 80$ $h = 80$

Along the curve

③

Response curve — green

M_p=f_y Z=f_y\,bh^2/4

$h \in \left[80, 900\right]$

$h = 80$ $M_p = 113.6$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}M_p}{\mathrm{d}h}

$h \in \left[80, 900\right]$

$h = 85.16$ $M_p^{\prime} = 3.023$

Along the curve

Lecture brief

Historical brief

From Euler’s 1744 elastica and Navier’s beam theory to Mohr’s circle and transformed-section RC, structural mechanics grew as a closed-form craft before finite elements. The lab keeps those governing lines for buckling, flexure, joints and influence. This sheet (CIV-22 — Plastic moment Mp) is the form associated with Plastic modulus Z. Working symbols: $f_y$ , $b$ , $h$ $\rightarrow$ $M_p$ . The whole section yields. Shape factor f = Z/S is 1.5 for a rectangle, ≈1.12 for an I-beam.

Purpose

Purpose: compute

M_p

from

f_y

b

h

in Structural & civil via

M_p=f_y Z=f_y\,bh^2/4

Mp = fy Z. For a rectangle Z = bh²/4 so Mp = fy b h²/4. Use it when a real structural & civil question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

f_y = 355.000\,\mathrm{MPa}

b = 200.000\,\mathrm{mm}

h = 400.000\,\mathrm{mm}

, the governing relation

M_p=f_y Z=f_y\,bh^2/4

yields

M_p = 2840.0\,\mathrm{kN·m}

. A beam, a fully plastic hinge, a moment plateau. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Yield stress f_y355 MPaWidth b200 mmDepth h400 mm

Outputs

Plastic moment M_p2840.0 kN·m

Reading speed

Watch on YouTube

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Narration of this film

A beam, a fully plastic hinge, a moment plateau.

The whole section yields. Shape factor f = Z/S is 1.5 for a rectangle, ≈1.12 for an I-beam.

Reading speed

Watch on YouTube