CIV-02

Euler–Bernoulli flexure formula

Bending stress σ = My/I in a slender beam.

Reading speed

FlexureNavier 1826EN 1992-1-1

Governing equation

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

$M \in \left[1, 2000\right]$

where

M: Bending moment (kN·m)
y: Fibre distance (mm)
I: Second moment (cm⁴)
\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}

$M \in \left[1, 2000\right]$

$M = 1$ $\sigma = 3$

Along the curve

②

Domain of the variable — blue

M_{\min} = 1,\quad M_{\max} = 2000

$M \in \left[1, 2000\right]$

$M = 1$ $M = 1$

Along the curve

③

Response curve — green

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

$M \in \left[1, 2000\right]$

$M = 1$ $\sigma = 3$

Along the curve

④

Rate of change — amber

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

$M \in \left[1, 2000\right]$

$M = 13.57$ $\sigma^{\prime} = 3$

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-02 — Euler–Bernoulli flexure formula) is the form associated with Navier 1826 · EN 1992-1-1. Working symbols: $M$ , $y$ , $I$ $\rightarrow$ $\sigma$ . Plane sections remain plane, so εx = −y/ρ. Hooke's law and equilibrium of moment give the Navier formula σ = My/I.

Purpose

Purpose: compute

\sigma

from

M

y

I

in Structural & civil via

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

Bending stress σ = My/I in a slender beam. 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 = 5000.000\,\mathrm{cm⁴}

, the governing relation

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

yields

\sigma = 240.000\,\mathrm{MPa}

. Pure bending of a prismatic elastic beam. Enter M, y, I. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Bending moment M80 kN·mFibre distance y150 mmSecond moment I5000 cm⁴

Outputs

Bending stress \sigma240.000 MPa

Reading speed

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Narration of this film

Pure bending of a prismatic elastic beam. Enter M, y, I.

Plane sections remain plane, so εx = −y/ρ. Hooke's law and equilibrium of moment give the Navier formula σ = My/I.

Reading speed

Watch on YouTube