CIV-12

Plastic shape factor

f = Z/W; rectangle gives f = 1.5.

Reading speed

PlasticityEN 1993-1-1AISC 360

Governing equation

W=\dfrac{b h^2}{6},\quad Z=\dfrac{b h^2}{4},\quad f=\dfrac{Z}{W}=\dfrac32

$h \in \left[40, 1200\right]$

where

b: Width (mm)
h: Depth (mm)
f_y: Yield stress (MPa)
W: Elastic modulus (cm³)
Z: Plastic modulus (cm³)
f: Shape factor (—)
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

W=\dfrac{b h^2}{6},\quad Z=\dfrac{b h^2}{4},\quad f=\dfrac{Z}{W}=\dfrac32

$h \in \left[40, 1200\right]$

$h = 40$ $W = 28400$

Along the curve

②

Domain of the variable — blue

h_{\min} = 40,\quad h_{\max} = 1200

$h \in \left[40, 1200\right]$

$h = 40$ $h = 40$

Along the curve

③

Response curve — green

W=\dfrac{b h^2}{6},\quad Z=\dfrac{b h^2}{4},\quad f=\dfrac{Z}{W}=\dfrac32

$h \in \left[40, 1200\right]$

$h = 40$ $W = 28400$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}W}{\mathrm{d}h}

$h \in \left[40, 1200\right]$

$h = 47.3$ $W^{\prime} = 1679$

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-12 — Plastic shape factor) is the form associated with EN 1993-1-1 · AISC 360. Working symbols: $b$ , $h$ , $f_y$ $\rightarrow$ $W$ , $Z$ , $f$ , $M_p$ . The plastic modulus Z is the first moment of area about the plastic neutral axis. Shape factor f = Mp/My = Z/W measures reserve beyond first yield.

Purpose

Purpose: compute

W

Z

f

M_p

from

b

h

f_y

in Structural & civil via

W=\dfrac{b h^2}{6},\quad Z=\dfrac{b h^2}{4},\quad f=\dfrac{Z}{W}=\dfrac32

f = Z/W; rectangle gives f = 1.5. 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

b = 200.000\,\mathrm{mm}

h = 400.000\,\mathrm{mm}

f_y = 355.000\,\mathrm{MPa}

, the governing relation

W=\dfrac{b h^2}{6},\quad Z=\dfrac{b h^2}{4},\quad f=\dfrac{Z}{W}=\dfrac32

yields

W = 5333.333\,\mathrm{cm^{3}}

Z = 8000.000\,\mathrm{cm^{3}}

f = 1.500\,\mathrm{—}

M_p = 2.840e+6\,\mathrm{kN·m}

. Rectangle b×h, elastic-perfectly plastic steel. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Width b200 mmDepth h400 mmYield stress f_y355 MPa

Outputs

Elastic modulus W5333.333 cm³
Plastic modulus Z8000.000 cm³
Shape factor f1.500 —
Plastic moment M_p2840000.000 kN·m

Reading speed

Watch on YouTube

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Narration of this film

Rectangle b×h, elastic-perfectly plastic steel.

The plastic modulus Z is the first moment of area about the plastic neutral axis. Shape factor f = Mp/My = Z/W measures reserve beyond first yield.

Reading speed

Watch on YouTube