CIV-14

Elastic flexure stress

σ = M y / I. Navier's hypothesis: plane sections remain plane.

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FlexureEuler–Bernoulli

Governing equation

\sigma=\dfrac{M y}{I}

$M \in \left[5, 400\right]$

where

M: Moment (kN·m)
y: Fibre distance (mm)
I: Second moment (cm^4)
\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

\sigma=\dfrac{M y}{I}

$M \in \left[5, 400\right]$

$M = 5$ $\sigma = 6250$

Along the curve

②

Domain of the variable — blue

M_{\min} = 5,\quad M_{\max} = 400

$M \in \left[5, 400\right]$

$M = 5$ $M = 5$

Along the curve

③

Response curve — green

\sigma=\dfrac{M y}{I}

$M \in \left[5, 400\right]$

$M = 5$ $\sigma = 6250$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}\sigma}{\mathrm{d}M}

$M \in \left[5, 400\right]$

$M = 7.484$ $\sigma^{\prime} = 1250$

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-14 — Elastic flexure stress) is the form associated with Euler–Bernoulli. Working symbols: $M$ , $y$ , $I$ $\rightarrow$ $\sigma$ . The curvature κ = M/EI; fibre strain is κ y; Hooke maps strain to stress.

Purpose

Purpose: compute

\sigma

from

M

y

I

in Structural & civil via

\sigma=\dfrac{M y}{I}

σ = M y / I. Navier's hypothesis: plane sections remain plane. 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

M = 80.000\,\mathrm{kN·m}

y = 150.000\,\mathrm{mm}

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

, the governing relation

\sigma=\dfrac{M y}{I}

yields

\sigma = 100000.00\,\mathrm{MPa}

. A beam, a moment couple, fibres in tension and compression. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Moment M80 kN·mFibre distance y150 mmSecond moment I12000 cm^4

Outputs

Bending stress \sigma100000.00 MPa

Reading speed

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Narration of this film

A beam, a moment couple, fibres in tension and compression.

The curvature κ = M/EI; fibre strain is κ y; Hooke maps strain to stress.

Reading speed

Watch on YouTube