Neutral axis: Definition, Diagram, Formula, For different shapes [with Pdf]

What is neutral axis?

neutral axis diagram

Neutral axis for the beam subjected to bending is a line passing through the cross-section at which the fibres of the beam does not experience any longitudinal stress (compressive or tensile).

For the above beam, the dotted line N.A. indicates the neutral axis.

As shown in the above figure, due to the bending moment on the beam, the fibres above the neutral axis are subjected to compression and the fibres below the neutral axis are subjected to tension.

The uppermost fibres of the beam are under higher compressive stress and this compressive bending stress gradually decreases as we move towards the neutral axis and it becomes zero at the neutral axis.

Similarly, the bottommost fibre of the beam experiences the highest tensile stress and it gradually decreases as we move toward the neutral axis and it becomes zero at the neutral axis.

Therefore the fibres at the neutral axis are not subjected to compression as well as tension.

And thus it becomes clear that at the neutral axis the value of bending stress is zero.

How to find neutral axis?

The method to find a position of the neutral axis is different for simple beam and composite beam:-

A] Neutral axis of a simple beam:-

For a simple beam, the neutral axis passes through the centroid of the cross-sectional area. Therefore it can be easily found by finding the position of the centroid in a vertical direction.

neutral axis of simple beam

The position of the neutral axis is given by,

`\bar{y}=\frac{\sum A_{i}y_{i}}{\sum A_{i}}`

B] Neutral axis of composite beam:-

The composite beam, consist of a different material. The neutral axis of the composite section passes through the centroid of an equivalent cross-section. To find the neutral axis of such composite beam it is necessary to convert the actual cross-section into the equivalent section with the same modulus of elasticity and then find the centroid of the equivalent cross-section.

Step 1:- Convert composite cross-section into equivalent cross-section

neutral axis of composite beam

As shown in Fig. A, The cross-section consists of a material with a modulus of elasticity `E_{1}` and `E_{2}`.

To convert the composite section into the equivalent cross-section with an equivalent modulus of elastic of `E_{1}`.

Multiply the width of section 2 by `n_{2}`.

Where, `n_{2}` = `\frac{E_{2}}{E_{1}}`

Therefore new width of section 2 is,

`b_{2}` = `n_{1}.b`

Multiply the width of section 3 by `n_{3}`,

Where, `n_{3}` = `\frac{E_{3}}{E_{1}}`

Therefore new width of section 3 is,

∴ `b_{3}` = `n_{3}` .b

Step 2:- Find position of the neutral axis

Now by simply finding the centroid of a composite section in a vertical direction, we can find the position of the neutral axis.

neutral axis of composite beams

`\bar{y}=\frac{\sum A_{i}y_{i}}{\sum A_{i}}`

Where, `\bar{y}_{\text{Composite}}=\bar{y}_{\text{Equivalent}}`

Neutral axis equation for geometric shapes:

1] Neutral axis of T beam:-

neutral axis of t beam

The position of the neutral axis from the bottom of the web is given by,

`\bar{y}`=`\frac{A_{\text{web}}.y_{\text{web}}+A_{\text{flange}}.y_{\text{flange}}}{A_{\text{web}}+A_{\text{flange}}}`

Where,
`A_{\text{Web}}` = Area of web
`A_{\text{Flange}}`= Area of flange
`y_{\text{Web}}`= Centroid of the web from bottom of web
`A_{\text{Flange}}`= Centroid of the flange from bottom of web

2] Neutral axis of a circle:-

neutral axis of circle

For the circular section, the centroid is at the centre of the circle therefore the neutral axis of the circle is given by,

`\bar{y}=\frac{d}{2}`

3] Neutral axis of rectangle:-

neutral axis of rectangle

For the rectangular cross-section, the neutral axis passes through the centroid.

Therefore the position of the neutral axis for the rectangle is given by,

`\bar{y}=\frac{d}{2}`

4] Neutral axis of triangle:-

neutral axis of triangle

For the equilateral triangular cross-section, the neutral axis passes through the centroid.

Therefore the position of the neutral axis for the equilateral triangle is given by,

`\bar{y}=\frac{h}{3}`

Why bending stress is zero at neutral axis?

For the beam subjected to the bending, the outermost fibre at one end experience the highest tensile stress and the outermost fibres at the opposite end experiences the highest compressive stress.

Therefore for the cross-section of the beam, all the fibres from one of the outermost ends to the opposite outermost end are subjected to the varying bending stress from highest tensile stress to the highest compressive stress.

The bending stress from the highest tensile stress gradually decreases to zero and again gradually increases to the highest compressive stress.

At the certain location where the bending stress changes its nature from tensile to compressive, the value of the bending stress becomes zero and thus the fibres at this location never experience any bending stress. This position is known as the neutral axis.

Therefore at the neutral axis, the value of the bending stress is zero.

FAQ’s

  1. What do you meant by neutral axis?

    A neutral axis is a line passing through the cross-section at which the fibres of the beam does not experience any longitudinal stress

  2. How do you find the neutral axis?

    The neutral axis passes through the centroid of the cross-section, therefore by finding the location of the centroid we can locate the neutral axis.

  3. What is neutral axis and neutral plane?

    The neutral axis is a line passing through the cross-section at which the fibres of the beam do not experience any longitudinal stress and the neutral plane is the plane in the beam at which all the fibres do not experience any longitudinal stress.

  4. Where is neutral axis of a beam?

    The elastic neutral axis passes through the centroid of the beam cross-section.

  5. Does neutral axis always pass through centroid?

    The elastic neutral axis always passes through the centroid of the cross-section and the plastic neutral axis pass through the line that divides the cross-sectional area into two parts of equal area.

  6. What will be bending stress at neutral axis?

    The bending stress at the neutral axis is zero since at the neutral axis the bending stress changes its nature from compressive to tensile.

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