# Moments of Area

First and Second Moment of Area

## Moment in Physics

In Physics Moment (or Torque) is **force times distance**:

But there are other Moments, read on!

## First Moment of Area

First Moment of Area is **area times distance** (to some reference line):

First Moment of Area = A x d

For this simple case we can multiply the whole area by the distance (from its middle to the reference line).

### Example:

Area is 20 mm x 10 mm = 200 mm^{2}

First Moment of Area (relative to the bottom line) = 200 mm^{2} x 25 mm = 5000 mm^{3}

(Note the unit is mm^{3}, but is not a volume!)

One of it's great uses is to find the centroid , which is the average position of all the points of an object:

A plane shape cut from a piece of card will balance perfectly on its centroid.

To find the distance to the centroid from any axis, we divide the First Moment of Area by the Total Area:

When we do that for both the x-axis and y-axis we get the centroid.

We can **estimate** where the centroid is using squares:

## Second Moment of Area

For the **Second Moment of Area** we multiply the area by the distance **squared**:

(need infinitely many tiny squares)

But be careful! We need to multiply every tiny bit of area by its distance squared, because area further away has a bigger effect (due to the distance being squared).

It is called "Second" moment because we square the distance "x^{2}"

It is also called the **area moment of inertia**.

We can **estimate** the second moment using squares, but it is very inaccurate:

We can use the x-axis or y-axis as the reference line, or we can use the **centroid for the reference line**. You can try that option above.

**Notation**: The symbol is an "I" followed by a little "x" or "y" for the reference axis.

I_{x} is in relation to the **x** axis (and we use **y** distances times area)

I_{y} is in relation to the **y** axis (and we use **x** distances times area)

The letter **I** refers to **Inertia** in "area moment of inertia".

Where possible use an accurate formula such as:

_{x}= \frac{bh^{3}}{3}

I

_{y}= \frac{b^{3}h}{3}

_{x}= \frac{bh^{3}}{12}

I

_{y}= \frac{b^{3}h}{12}

_{x}= \frac{bh^{3}}{12}

I

_{y}= \frac{b^{3}h+b^{2}ha+bha^{2}}{12}

_{x}= \frac{πr^{4}}{4}

I

_{y}= \frac{πr^{4}}{4}

Engineers use the second moment of area to work out how rigid (hard to bend) a beam is.

### Example: A beam that is 100 mm by 24 mm

Lying flat it looks like this:

I_{x} = \frac{bh^{3}}{12} = \frac{100 × 24^{3}}{12} = **115,200** mm^{4}

But sitting upright it is:

I_{x} = \frac{bh^{3}}{12} = \frac{24 × 100^{3}}{12} = **2,000,000** mm^{4}

It is nearly 20 times as rigid sitting upright!

And that is why beams sit up like this:

Try bending a ruler about each axis to experience it for yourself:

## Engineers Love I-Beams

Here we have two equal-sized beams, but one is solid, the other shaped like an "I"

The solid beam is a bit stiffer against bending (I_{x} = 333 vs 205) but **very much heavier** (40kg vs 16kg).

In practice we could have a slightly bigger I-Beam and still save a lot of money in steel, transport and handling.

To calculate the second moment of area for an I-Beam we can break it down into rectangles and get these formulas:

_{x}= \frac{wh^{3} − (w−b)(h−2t)^{3}}{12}

I

_{y}= \frac{w^{3}h − (w−b)^{3}(h−2t)}{12}

But **in practice** it is best to use the manufacturer's tables of beam properties as they account for any odd shapes.