First moment of area

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 the 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}`

About y-axis,

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

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 X and Y-axis is given by,

dQ🇾​​​​​ = 𝓧. dA
dQx = 𝓨.dA

Thus the first moment of area of total shape about X and Y-axis can be calculated as,

Q🇾​​​​​ = ∫ 𝓧.dA
Qx = ∫ 𝓨.dA

First moment of area 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 distance between the centroid of 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

First moment of area of different shapes:

The first moment of area for some shapes is given below:-

1] First moment of area of circle:-

A] First moment of area about axis away from centroid:

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

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

`Q_{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

B] First moment of area of either side about a centroidal axis:

First moment of area of circle about centroidal axis

The first moment of area of an upper half-circle about a centroidal axis is given by,

`Q_{C}` = `A_{\text{Upper}}` 𝖷 𝓨

`Q_{C}` = `(\frac{\pi r^{2}}{2}) \times \frac{4r}{3\pi}`

`Q_{C}` = `\frac{2}{3}r^{3}`
2] First moment of area of Hollow circle:-

A] First moment of area about axis away from centroid:

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

b] Fist moment of area of either side about a centroidal axis:-

First moment of area of hollow circle about centroidal axis

Thus the first moment of area of upper half part about a centroidal axis is given by,

`Q_{C}` = `[A_{o} \times y_{o}] – [A_{i} \times y_{i}]`

`Q_{C}` = `[\frac{\pi.r_{o}^{2}}{2} \times\frac{4r_{o}}{3\pi}]` – `[\frac{\pi.r_{i}^{2}}{2} \times\frac{4r_{i}}{3\pi}]`

`Q_{C}` = `\frac{2}{3}r_{o}^{3}` – `\frac{2}{3}r_{i}^{3}`

`Q_{C}` = `\frac{2}{3}[r_{o}^{3} – r_{i}^{3}]`
3] First moment of area of Rectangle:-

A] First moment of area about axis away from centroid:-

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

b] First moment of area of either side about a centroidal axis:-

First moment of area of rectangle about centroidal axis

Thus the first moment of area of upper half about a centroidal axis is given by,

`Q_{C}` = A x 𝓨

`Q_{C}` = `(\frac{d}{2} \times b) \times \frac{d}{4}`

`Q_{C}` = `\frac{bd^{2}}{8}`

First moment of area example:

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.

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
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

Position of centroid of each shape from 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 cm³

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 x 1 = 3 cm²
`A_{2}` = 1 x 0.5 = 0.5 cm²

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

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

Now the first moment of area of 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 cm³

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. What is first moment of area formula?

    The formula of the first moment of area (Q) about reference axis (𝓧) is,
    Q = Σ A𝔦.𝓨𝔦
    Where,
    A = Area of each elemental shape
    𝓨i = Distance between the centroid of each elemental shape from the reference axis.

  3. 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).

  4. 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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