First Moment of Area
In this post, we will be learning why and how to calculate the first moment of area. To do this, we will first look at the equations that need to be used, break them down and understand the variables. From this, we will cover a basic example to get our feet wet. Once all this is clear, we will proceed to work through a more complex shape. Before going ahead with this post, we recommend that you have a good understanding of how to calculate the centroid. The post here explains with worked examples how to find the centroid.
What is the First Moment of Area?
The first moment of area is the measurement of the beam’s sectional area relative to an axis. E.g. X or Y in typical Cartesian co-ordinates. It can be used to calculate the centroid. This will be when the sum of the first moments of area are equal from a datum or reference point.
How Do We Use the First Moment of Area?
Now we know what the first moment of area actually is (well kind of, the theory seldom makes sense until you go through the process of actually calculating it). Does it have a use? Yes, it is the ratio of the section in relation to the area, which enables you to calculate parameters at certain parts in the section. These include the neutral axis location (also known as the centroid) and the first moment of area, which will be at a maximum (or equal).
The other useful application of calculating the first moment of area is that you can calculate the neutral axis of the section, or the centroid of the shape. This will be explained during the examples later on, but have a think with the information above to see if you can figure out how!
How Do We Calculate the First Moment of Area?
Now for the fun part. Equations exist for basic shapes, such as squares and rectangles (which make up the vast majority of what we use in structural engineering). The first moment of area can also be calculated by integration, which is much better to understand since it is applicable to any geometric shapes. To save this from being a maths lesson, we will do a separate post on how to derive the main equations that we need to use. By putting this in a separate, more detailed post, it will not only help your understanding, but will also stop you from being bogged down if you are in a rush today. Keep an eye out for it!
Here is the breakdown of the variables in the equation for first moment of area for the X-Axis:
Qx = ∑(Yi*Ai)
or
Qy = ∑(Xi*Ai)
Yi = The distance from the datum or reference axis to the centre of the shape i.
Ai = The area of shape i.
If we wanted to find the maximum area in relation to the axis for a set shape, Xi would be the distance from the datum to the centroid.
Calculating the First Moment of Area - Ex. 1
To keep it very consistent, we are going to use the same examples as we did in the previous example of the centroid. Below is our rectangle. Our aim is to find the first moment of area about the centroid. Hopefully we can all remember that the centroid is found at (1, 2.5).
First Moment of Area - Ex.1
Again, being only one shape, we will only have Y1 and A1. We will calculate it for the X-axis and do it twice (two shapes) to show what happens when we take the first moment of area about the neutral axis (the centroid).
Step one: break the shape down into sub-shapes about the point where we are calculating the first moment area of. In this instance, it is going to be the centroid.
X-Axis
Qx = Y1* A1
So we break down the shape and then we can calculate the variables.
First Moment of Area Ex.1 - Qx,bottom
Qx, bottom = The First Moment of Area!
Y1 = 2.5 / 2 = 1.25
A1 = 5 * 1 = 5
Putting it altogether..
Qx, bottom = 1.25 * 5
X̄ = 6.25
Now to check the top part, Qx, top.
First Moment of Area Ex.1 - Qx, Top
Qx, top = The First Moment of Area!
Y1 = 2.5 / 2 = 1.25
A1 = 5 * 1 = 5
Qx, top = 1.25 * 5
= 6.25
We can see they are both identical. Therefore we have an equal measurement of area on both sides about the neutral axis.
Y-Axis
For the Y-axis, similar to the X-axis we break down the shape into two parts. For this case we will call it east and west.
Qy, east = X1 * A1
First Moment of Area Ex.1 - Qy, east
Qy, east = First moment of area
X1 = 1 / 2 = 0.5
A1 = 5 * 1 = 5
Qy, east = 0.5 * 5
= 2.5
And now for the other side, Qy, west.
First Moment of Area Ex.1 - Qy, west
Qy, west = First moment of area
X1 = 1 / 2 = 0.5
A1 = 5 * 1 = 5
Qy, west = 0.5 * 5
= 2.5
Again, since the first moment of area in the Y-axis is taken about the centroid, the areas are equal. This is a great way to check the answer is correct, especially if you’re in an exam!
**Calculating the First Moment of Area - Ex. 2**
Below is a composite shape made up of a square and a rectangle, seen previously in the centroid tutorial. We will now calculate the first moment of area about the neutral axis. If you remember, this was (1.25, 1.25).
First Moment of Area Ex.2
With composite shapes as above, we need to use a similar approach to that used when calculating the centroid. We will split the shape about the neutral axis, giving the potential to have multiple shapes. As before, we will separate the X and Y-Axis. In structural design, this could be known more commonly as the major and minor axis, but we will cover that in a separate topic.
X-Axis - Below the Centroid
We will start with the X axis first. It’s a little bit easier to split into sub-shapes than the centroid. As long as the shape is divided about the axis, it does not matter what sub-shape you have.
Overall shape below the Neutral Axis - Ex. 2
We now have the area of the shape we want to calculate the first moment of area for. Next we must split the shape into rectangles and then apply the formula. You can see the two shapes below, labelled as 1 and 2. Note that the hatching has been removed for clarity.
Two sub-shapes for Qx, Bottom - Ex.2
Now using the original equation, we can split Yi and Ai for both shapes. Recall that Y1 is the distance from the centroid of shape 1 to the neutral axis.
Qx,Bottom = ∑(Yi*Ai)
or
Qx,Bottom = (Y1 * A1) + (Y2 * A2)
Qx,Bottom = First Moment of Area
Y1 = 1 / 2 + (1.25 - 1) = 0.75
A1 = 2 * 1 = 2
Y2 = (1.25 - 1) / 2 = 0.125
A2 = (1.25 - 1) * 1 = 0.25
Qx, Bottom = (0.75 * 2) + (0.125 * 0.25)
Qx, Bottom = 1.53
X-Axis - Above the Centroid
For Qx, Top, it’s a basic rectangle which makes the whole process as simple as example 1.
Overall shape above the neutral Axis - Ex. 2
Qx,Top = First Moment of Area
Y1 = (3 - 1.25) / 2 = 0.875
A1 = (3 - 1.25) * 1 = 1.75
Qx, Top = (0.875 * 1.75)
Qx, Top = 1.53
Y-Axis - East of the Centroid
Now we must complete the process again, but for the Y-axis. This involves the distance Xi. We will have split the shapes up in the same manner as before, but it will be different.
Overall shape, east of the neutral axis - Ex. 2
And again, splitting the shape into sub-shapes and labeling them 1 and 2.
Shape split into two parts - Ex. 2
Qy, east = First Moment of Area
Y1 = (1 / 2) + 0.25 = 0.75
A1 = 1 * 1 = 1
Y2 = (0.25 / 2) = 0.125
A2 = (1.25 - 1) * 3 = 0.75
Qy, east = (0.75 * 1) + (0.125 * 0.75)
ȳ = 0.84
Y-Axis - West of the Centroid
Overall shape, west of the neutral axis - Ex. 2
Qy, west = First Moment of Area
Y1 = (2 - 1.25) / 2 = 0.375
A1 = (2 - 1.25) * 3 = 2.25
Qy, west = (0.375 * 2.25)
Qy, west = 0.84
So there we have it. We have the solution to both the first moment of area for the X-axis, Qx, and the Y-axis, Qy. On top of this, we have managed to check the answers we have are correct and we have therefore not made any simple mistakes in computing it. Obviously, if you are confident in the method to calculate the first moment of area, you would only need to do Qx and Qy once and would chose the easier of the shapes.
In summary,
Qx = 1.53
Qy, = 0.84
This would state that the shape has more area in relation to the X-axis and therefore the major axis of the shape is 1.53. Being the major axis means the properties are greater. This implies that the shape would have more capacity to resist forces in such a direction, assuming that the shape is uniform.