First moment of area: Definition, Formula, Units, Examples

What is First moment of area?

First moment of area about any reference axis is the product of the area of shape and distance between the centroid of shape and the reference axis.

The first moment of area is generally denoted by the symbol ‘Q’ and it has the dimensional formula of [L³ M⁰ T⁰].

First moment of area

For the shape shown in the above figure, the first moment of area about the x-axis is given by,

`Q_{x}` = A x 𝓨

Similarly, the first moment of area about the y-axis is given by,

`Q_{y}` = A x 𝓧

For a complex shape consisting of more than one simple geometric shape, the first moment of area is the summation of the product of the area of each section and the distance between its centroid from the reference axis.

Mathematically it is given by,

About x-axis,

`Q_{x}= \sum A_{i}y_{i}=A_{1}y_{1}+A_{2}y_{2}+\cdots + A_{n}y_{n}`

Where,
`A_{1}, A_{2}, A_{3}, \cdotsA_{n}` = Area of the each shape
`y_{1}, y_{2}, y_{3}, \cdots y_{n}` = Distance of the centroid of each shape from the x-axis

About y-axis,

`Q_{y}=\sum A_{i}x_{i}=A_{1}x_{1}+A_{2}x_{2}+\cdots+A_{n}x_{n}`

Where,
`x_{1}, x_{2}, x_{3}, \cdots x_{n}` = Distance of the centroid of each shape from the y-axis

First moment of area formula:

The first moment of area of the complex shape about any reference axis (X and Y) is given by,

Qx = Σ A 𝓨
Q🇾​​​​​ = Σ A 𝓧

Where,
A = Area of each shape
𝓧 or 𝓨 = Distance between the centroid of shape and reference axis (X or Y).

First moment of area equations:

The below figure shows the plane lamina with an irregular shape.

first moment of area equation

Consider a small elemental area dA with the centroid located at coordinate (x, y).

By the definition of the first moment of area, the first moment of area of dA about the X and Y-axis is given by,

dQ🇾​​​​​ = 𝓧. dA
dQx = 𝓨.dA

Integrate the dQ🇾​​​​​ and dQx to find the first moment of area of total shape about the X and Y-axis,

Q🇾​​​​​ = ∫ 𝓧.dA
Qx = ∫ 𝓨.dA

Units:

SI unit:-

In the SI system, the unit of area is m² and the unit of distance is m.

∴ Q = A x distance = m² x m = m³

Thus the SI unit of the first moment of area is m³.

FPS unit:-

In the FPS system, the unit of area is ft² and the unit of distance is ft.

∴ Q = A x Distance = ft² x ft = ft³

Thus the FPS unit of the first moment of area is ft³.

How to calculate first moment of area?

Following are the steps to calculate the first moment of area of complex shapes:-

Step 1] Divide the complex shape into simple geometric shapes as shown below.

How to calculate first moment of area

Step 2] Find the distance between the centroid and reference axis for each shape (𝓧𝟭, 𝓧𝟮, 𝓧𝟯 or 𝓨𝟭, 𝓨𝟮, 𝓨𝟯).

Step 3] Find the area of each shape (A𝟭, A𝟮, A𝟯).

Step 4] Find the first moment of area using the following formulae,

For the above figure, the first moment of area about the 𝓧​​-axis is given by,

`Q_{X} =\sum A_{i}y_{i}`

`Q_{X}=A_{1}y_{1}+A_{2}y_{2} + A_{3}y_{3}`

For the above shape, the first moment of area about the 𝓨-axis is given by,

`Q_{Y}=\sum A_{i}x_{i}`

`Q_{Y}=A_{1}x_{1}+A_{2}x_{2}+A_{3}x_{3}`

When to use first moment of area?

The first moment of area is used for the following purpose:-

1] To find the centroid of complex shapes:-
For the complex shape consisting of different simple geometric shapes, the position of centroid from the X- axis (`\bar{Y}`) can be calculated as,

`\bar{Y}=\frac{\sum Q_{\text{xi}}}{A_{\text{Total}}}`

`\bar{Y}=\frac{Q_{x1} + Q_{x2} + \cdots + Q_{xn}}{A_{\text{Total}}}`

`\bar{Y}=\frac{A_{1}y_{1} + A_{2}y_{2} + \cdots + A_{1}y_{n}}{A_{\text{Total}}}`

Where,
`Q_{x1}, Q_{x2}, \cdots, Q_{xn}` = First moment of area of each shape about the X-axis
`A_{\text{total}}` = Total area of complex shape

And the position of centroid from the Y-axis (`\bar{X}`) is given by,

`\bar{X}=\frac{\sum Q_{yi}}{A_{\text{Total}}}`

`\bar{X}=\frac{Q_{y1} + Q_{y2} + \cdots + Q_{yn}}{A_{\text{Total}}}`

`\bar{X}=\frac{A_{1}x_{1} + A_{2}x_{2} + \cdots + A_{1}x_{n}}{A_{\text{Total}}}`

Where,
`Q_{y1}, Q_{y2}, \cdots Q_{yn}` = First moment of area of each shape about the Y-axis
`A_{\text{total}}` = Total area of complex shape

2] For the object subjected to the bending load, the first moment of area is necessary to find the transverse shear stress.

When is first moment of area zero?

At the centroidal axis, the first moment of area of the object becomes zero.

The first moment of area is the product of the area of the shape and the distance between the centroid of the shape and the reference axis.

Q = A`\times`𝓧

The centroidal axis passes through the centroid of the shape.

∴ 𝓧 = 0

Therefore the first moment of area at the centroidal axis is,

Q = A x 0

∴ Q = 0

For different shapes:

The first moment of area for circle, hollow circle, and rectangle shape is given below:-

1] For circle:-

The below figure shows the circle with the centroid located at a distance of (x, y) from the origin of the axis.

First moment of area of circle

For the above circle, the first moment of area about the 𝓧-axis is given by,

`Q_{x}` = A x 𝓨

As, area of the circle, A = π.r²

`\thereforeQ_{x}` = π.r² 𝓨

Similarly, the first moment of area about the 𝓨-axis is given by,

`Q_{Y}`​​​​​ = A x 𝓧

`Q_{Y}`​​​​​ = πr² 𝓧

2] For Hollow circle:-

The below figure shows the hollow circular shape with the centroid located at a distance of (x, y) from the origin.

First moment of area of hollow circle

For the above hollow circle, the first moment of area about the 𝓧-axis is given by,

`Q_{x}` = A x 𝓨

`Q_{x}=(\pi.r_{o}^{2} – \pi.r_{i}^{2})` x 𝓨

`Q_{x}=\pi.(r_{o}^{2} – r_{i}^{2})` x 𝓨

Similarly, the first moment of area of a circle about the 𝓨-axis is given by,

`Q_{y}` = A x 𝓧

`Q_{y}=(\pi.r_{o}^{2} – \pi.r_{i}^{2})` x 𝓧

`Q_{y}=\pi.(r_{o}^{2} – r_{i}^{2})` x 𝓧

3] For Rectangle:-

The below figure shows the rectangle of width b and height d with the centroid located at a distance of (x, y) from the origin.

First moment of area of rectangle

For an above rectangle, the first moment of area about the 𝓧-axis is given by,

`Q_{x}` = A x 𝓨

`Q_{x}` = bd x 𝓨

Similarly, the first moment of area of a rectangle about the 𝓨-axis is given by,

`Q_{y}` = A x 𝓧

`Q_{y}` = bd x 𝓧

First moment of area solved examples:

For the I beam shown below, find the first moment of area about the 𝓧-axis and the first moment of area of half-section about the neutral axis of beam.

First moment of area example

Solution:-

1] First moment of area about x-axis:-

First moment of area example 1

Divide the I-beam into simple geometric shapes
The area of each shape is given by,
`A_{1}` = 3 x 1 = 3 cm²
`A_{2}` = 1 x 1 = 1 cm²
`A_{3}` = 3 x 1 = 3 cm²

The position of the centroid of each shape from the x-axis is given by,
`y_{1}` = 2.5 cm
`y_{2}` = 1.5 cm
`y_{3}` = 0.5 cm

Now the first moment of area about the x-axis is given by,

`Q_{x}` = `\sum A_{i}y_{i}`

`Q_{x}` = `A_{1}y_{1}` + `A_{2}y_{2}` + `A_{3}y_{3}`

`Q_{x}` = (3 x 2.5) + (1 x 1.5) + (3 x 0.5)

`\mathbf{Q_{x}}=10.5\ \text(cm)^{3}`

2] First moment of area of half-section about a neutral axis:-

Given I-beam is symmetric thus the neutral axis passes through the centre of the I-beam.

First moment of area example 2

Now the area of each shape of the section is,
`A_{1}= 3 \times 1 = 3\ \text{cm}^{2}`
`A_{2}= 1 \times 0.5 = 0.5\ \text{cm}^{2}`

The distance between centroid and neutral axis for each shape is given by,

`y_{1}=1\ \text{cm}`
`y_{2}=0.25\ \text{cm}`

Now the first moment of area of the upper half-section about a neutral axis is given by,

`Q_{C}=\sum A_{i}.y_{i}`

`Q_{C}=A_{1}y_{1} + A_{2}y_{2}`

`Q_{C}` = (3 x 1) + (0.5 x 0.25)

`\mathbf{Q_{C}}=3.125\ \text{cm}^{3}`

FAQs:

  1. What is meant by first moment of area?

    The first moment of area of shape about any reference axis is the product of area and distance between the centroid of shape and reference axis.

  2. How do you find the first moment of the area of a circle?

    The first moment of area of a circle can be found by,
    Q = (Area of a circle) x (Distance between centroid and reference axis).

  3. How do you find the first moment of a rectangle given the area?

    The first moment of area of a rectangle can be found by,
    Q = (Area of a rectangle) x (Distance between centroid and reference axis).

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