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Created: 2024/05/01 17:29

Bending stress

It is the measure of internal pressure within a material that resists any change in shape when a bending moment is applied.

Mathematically, it is expressed as:

$$\sigma =\frac{M}{S}$$

Where,

• $M$ is the bending moment,
• $S$ is the section modulus.

In turn, $S$ section modulus is the measure of the material's capacity to resist bending. Mathematically, it is the ratio between second moment of inertia and the distance from neutral axis to the point of maximum stress.

$$S=\frac{I}{y}$$

Hence, bending stress can be re-written as:

$$\sigma =\frac{My}{I}$$

[!Note] Neutral axis (NA) The axis along which bending causes neither compression nor tension is called neutral axis.

Bending stress in beam

Any weights applied to a beam induces a bending moment and generates a compressional stress on one side of the beam, and an equal tensile stress on the other side of the beam.

It is assumed that plane section remains plane after bending. That means, any section that is perpendicular to the neutral axis, remains perpendicular (to that bended axis) even after bending. This assumption is taken to simplify the theory as there is no shear stress to deal with.

This stress can be calculated using:

$$\sigma =\frac{My}{I}$$

[!Note] Note For rectangles, no stress is observed through the centroidal axis.

Theoretical deflection

For four point bending, theoretical maximum deflection can be calculated using:

$${\delta }_{max}=\frac{23P{L}^3}{648EI}$$

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References:

• Bending Stress: Examples, Applications & Formula (studysmarter.co.uk)
• https://qr.ae/pswiYV