Bending stress: the flexure formula

By the end of this topic you will be able to

Locate the neutral axis and describe the bending stress distribution across a section.
Apply σ = My/I and σmax = M/Z.
Compare rectangular, circular and I-sections for efficiency.

How a beam bends

When a beam sags, the top fibres shorten and the bottom fibres lengthen. Somewhere between them lies a layer that keeps its length: the neutral axis, which passes through the centroid of the cross-section for a linear-elastic beam.

Strain grows in proportion to the distance $y$ from the neutral axis, and so does stress (Hooke's law). That gives the flexure formula:

\sigma = \frac{My}{I}

where $M$ is the bending moment at the section, $I$ is the second moment of area about the neutral axis, and $y$ is measured from the neutral axis. The stress is greatest at the extreme fibre $y = c$ :

\sigma_{\text{max}} = \frac{Mc}{I} = \frac{M}{Z}, \qquad Z = \frac{I}{c}

$Z$ is the section modulus. For a rectangle $Z = bh^2/6$ . Manufacturers' tables list $Z$ for standard beams so that you can pick a section quickly.

Compression above, tension below

For a sagging moment the top half is in compression and the bottom half in tension, with zero stress at the neutral axis. The stress diagram is two triangles, and the same maximum stress occurs top and bottom if the section is symmetric.

Why an I-section?

The material near the neutral axis carries little stress. An I-section puts most of its area in the flanges, far from the axis, where it does most of the work. For the same area, it has a much larger $I$ , so it can carry a larger moment, or the same moment with less steel. In the model, compare a rectangle with an I-section: the I-section gets a far higher I per unit area.

Interactive modelChoose a section, apply a moment, and read the stress at any level.

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Worked example

A timber beam of rectangular section 80 mm × 200 mm carries a maximum bending moment of 40 kN·m. Find the maximum stress and the stress 50 mm above the neutral axis.

Section properties. $I = 80\times200^3/12 = 53.33\times10^6$ mm⁴ and $c = 100$ mm, so $Z = I/c = 533\,300$ mm³.

Maximum stress. $\sigma_{\text{max}} = \dfrac{M}{Z} = \dfrac{40\times10^6}{533\,300} = 75.0$ MPa (compression at the top, tension at the bottom).

At $y = 50$ mm. $\sigma = \dfrac{40\times10^6\times50}{53.33\times10^6} = 37.5$ MPa. Halfway to the extreme fibre gives half the stress.

Practice problems

A 100 mm wide, 150 mm deep rectangular beam carries $M = 12$ kN·m. Find $\sigma_{\text{max}}$ .

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$Z = 100\times150^2/6 = 375\,000$ mm³. $\sigma = 12\times10^6/375\,000 = 32$ MPa.
The same beam is turned on its side (150 mm wide, 100 mm deep). Find $\sigma_{\text{max}}$ for the same moment.

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$Z = 150\times100^2/6 = 250\,000$ mm³, so $\sigma = 48$ MPa, 50% higher. Always place the long side vertical.
The allowable stress is 10 MPa for a timber beam carrying 8 kN·m. Find the minimum section modulus.

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$Z = M/\sigma = 8\times10^6/10 = 800\,000$ mm³.
Compare a 100 × 100 mm square section with an I-section of 100 mm flanges, 12 mm flange thickness, 8 mm web and 200 mm overall depth. Which has the larger $I$ and which has the larger area?

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Square: $A = 10\,000$ mm², $I = 8.33\times10^6$ mm⁴. I-section: $A = 3808$ mm², $I = 24.9\times10^6$ mm⁴. The I-section has about three times the $I$ with only 38% of the area (try it in the model).

Key formulae

Flexure formula: $\sigma = \frac{My}{I}$
Maximum bending stress: $\sigma_{\text{max}} = \frac{Mc}{I} = \frac{M}{Z}$
Section modulus: $Z = \frac{I}{c}$
Rectangular section: $Z = \frac{bh^2}{6}$

Lecturer notes: suggested plan, activity and assessment

Suggested plan (45 min). 10 min: bend a foam block with grid lines to show the neutral axis; 10 min: derive the flexure formula; 10 min: worked example; 15 min: sections in the model.

Activity. Ask students to find which section carries the largest moment within 100 MPa using roughly equal area.

Discussion. Why are tubes and I-sections efficient in bending but not in all cases (buckling of thin webs)?

Assessment idea. Design a timber floor joist for a given moment and allowable stress.