CIV-26

Elastic section modulus

S = I / c, σ = M / S. The extreme-fibre elastic stress.

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FlexureSection modulus

Governing equation

S=I/c,\quad \sigma=M/S

$M \in \left[1, 800\right]$

where

I: Second moment (cm^4)
c: Extreme fibre (mm)
M: Moment (kN·m)
S: Section modulus (cm^3)
\sigma: Bending stress (MPa)

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

S=I/c,\quad \sigma=M/S

$M \in \left[1, 800\right]$

$M = 1$ $S = 1250$

Along the curve

②

Domain of the variable — blue

M_{\min} = 1,\quad M_{\max} = 800

$M \in \left[1, 800\right]$

$M = 1$ $M = 1$

Along the curve

③

Response curve — green

S=I/c,\quad \sigma=M/S

$M \in \left[1, 800\right]$

$M = 1$ $S = 1250$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}S}{\mathrm{d}M}

$M \in \left[1, 800\right]$

$M = 6.025$ $S^{\prime} = 1250$

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-26 — Elastic section modulus) is the form associated with Section modulus. Working symbols: $I$ , $c$ , $M$ $\rightarrow$ $S$ , $\sigma$ . Navier: plane sections, linear strain, Hooke. S packages I and the farthest fibre.

Purpose

Purpose: compute

S

\sigma

from

I

c

M

in Structural & civil via

S=I/c,\quad \sigma=M/S

S = I / c, σ = M / S. The extreme-fibre elastic stress. 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

I = 12000.000\,\mathrm{cm^4}

c = 150.000\,\mathrm{mm}

M = 80.000\,\mathrm{kN·m}

, the governing relation

S=I/c,\quad \sigma=M/S

yields

S = 800.0\,\mathrm{cm^3}

\sigma = 100000.00\,\mathrm{MPa}

. A cross-section, a moment, an extreme fibre. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Second moment I12000 cm^4Extreme fibre c150 mmMoment M80 kN·m

Outputs

Section modulus S800.0 cm^3
Bending stress \sigma100000.00 MPa

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Narration of this film

A cross-section, a moment, an extreme fibre.

Navier: plane sections, linear strain, Hooke. S packages I and the farthest fibre.

Reading speed

Watch on YouTube