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

The first moment of area is just a product of the area of the lamina and the distance of its centroid from the axis. But it is difficult when the shape becomes complex. Sometimes, the first moment of area is incorrectly referred to as a first moment of inertia.

Contents

What is the 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. Finding the first moment of area is necessary to locate the centroid of the lamina and to find the transverse shear stress acting on the object.

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

For the shape shown in the above figure, the 1st moment of area about the x-axis is given by,

Qx = A x 𝓨

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

Qy = 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,

Qx = \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

Qy = \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 1st moment of area of the complex shape about any reference axis (X and Y) is given by,

Qx = Σ Ai 𝓨i
Qy​​​​​ = Σ Ai 𝓧i

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.

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

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

dQy = 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,

Qy​​​​​ = ∫ 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

∴ 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

∴ 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.

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,

Qx = \sum A_{i}y_{i}

Qx = 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,

Qy = \sum A_{i}x_{i}

Qy =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_{n}y_{n}}{A_{\text{Total}}}

Where,
Q_{x1}, Q_{x2}, \cdots, Q_{xn} = First moment of area of each shape about the X-axis
Atotal = 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_{n}x_{n}}{A_{\text{Total}}}

Where,
Q_{y1}, Q_{y2}, \cdots Q_{yn} = First moment of area of each shape about the Y-axis
Atotal = 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 first moment of area becomes 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\times𝓧

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.

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

Qx = 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,

Qy​​​ = A x 𝓧

Qy = 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.

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

Qx = A x 𝓨

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

Qx = \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,

Qy = A x 𝓧

Qy = (\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.

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

Qx = A x 𝓨

Qx = bd x 𝓨

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

Qy = A x 𝓧

Qy = 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.

Solution:-

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

Divide the I-beam into simple geometric shapes
The area of each shape is given by,
A1 = 3 x 1 = 3 cm²
A2 = 1 x 1 = 1 cm²
A3 = 3 x 1 = 3 cm²

The position of the centroid of each shape from the x-axis is given by,
y1 = 2.5 cm
y2 = 1.5 cm
y3 = 0.5 cm

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

Qx = \sum A_{i}y_{i}

Qx = A1y1 +A2y2 + A3y3

Qx = (3 x 2.5) + (1 x 1.5) + (3 x 0.5)

Qx = 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.

Now the area of each shape of the section is,
A1 = 3 × 1 = 3 cm2
A2 = 1 × 0.5 = 0.5 cm2

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

y1 = 1 cm
y2 = 0.25 cm

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

QC = \sum A_{i}y_{i}

QC = A1y1 + A2y2

QC = (3 x 1) + (0.5 x 0.25)

QC = 3.125 cm3

FAQs:

1. What is 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).

2. 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).

3. Where is the first moment of area zero?

The first moment of area is zero about the axis passing through the centroid.

4. When the first moment of area becomes negative?

It depends on the position of an axis. If the distance of the centroid of the shape from the axis is negative then the first moment of area also becomes negative.

Pratik is a Graduated Mechanical engineer. He enjoys sharing the engineering knowledge learned by him with people.