CIV-09

Combined axial plus bending

Extreme-fibre stress σ = P/A ± M y / I.

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Combined stressNavierEN 1993-1-1

Governing equation

\sigma = \dfrac{P}{A} \pm \dfrac{M y}{I}

where

P: Axial force (kN)
M: Moment (kN·m)
b: Width (mm)
h: Depth (mm)
\sigma_{\max}: Maximum stress (MPa)
\sigma_{\min}: Minimum stress (MPa)

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-09 — Combined axial plus bending) is the form associated with Navier · EN 1993-1-1. Working symbols: $P$ , $M$ , $b$ , $h$ $\rightarrow$ $\sigma_{\max}$ , $\sigma_{\min}$ . Superposition of uniform axial stress and Navier bending is valid while the section remains elastic and plane.

Purpose

Purpose: compute

\sigma_{\max}

\sigma_{\min}

from

P

M

b

h

in Structural & civil via

\sigma = \dfrac{P}{A} \pm \dfrac{M y}{I}

Extreme-fibre stress σ = P/A ± M y / I. 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

P = 400.000\,\mathrm{kN}

M = 60.000\,\mathrm{kN·m}

b = 300.000\,\mathrm{mm}

h = 500.000\,\mathrm{mm}

, the governing relation

\sigma = \dfrac{P}{A} \pm \dfrac{M y}{I}

yields

\sigma_{\max} = 7.467\,\mathrm{MPa}

\sigma_{\min} = -2.133\,\mathrm{MPa}

. Rectangular section, uniaxial moment about the strong axis. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Axial force P400 kNMoment M60 kN·mWidth b300 mmDepth h500 mm

Outputs

Maximum stress \sigma_{\max}7.467 MPa
Minimum stress \sigma_{\min}-2.133 MPa

Reading speed

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Narration of this film

Rectangular section, uniaxial moment about the strong axis.

Superposition of uniform axial stress and Navier bending is valid while the section remains elastic and plane.

Reading speed

Watch on YouTube