Welcome to the second part of the article on the analysis of cross-section parameters. In this section, I will cover everything you should know about geometric moments of inertia for cross-sections.
In this part of the article, we will cover the following topics:
Moment of inertia and deviation – how far are we from the center?
Assuming you’ve read the first part of the article, you should know that determining the location of the centroid of a cross-section involves certain quantities called static moments. You also know that their description can involve a lovely little integral…
As you can probably guess, physicists and mathematicians like to raise the bar, so we’ll start with moments of inertia in a similar way… But we’ll raise the power… preferably from the first to the second, just to make it manageable…
So let’s begin the preliminary description of moments of inertia by writing the general formula
I_x=∫_Ay^2 dA,
I_y=∫_Ax^2 dA
Now, seriously—these formulas are not some random creation of mathematicians. Specifically, these integrals result from solving the problem of the deflected axis of a bent beam, arising from the relationship between the moment acting on the beam and its radius of curvature ????. You could say that these integrals were derived from this solution and defined as geometric parameters later called geometric moments of inertia (second moment of area).
By analyzing the presented integrals, as before, we can say that in some way they describe the distribution of the cross-sectional area relative to the axes of a coordinate system. However, this time the coordinate is raised to the second power, meaning that moments of inertia cannot be negative!
Another piece of information we can infer from this integral is that the further the cross-sectional area is from the assumed axis, the greater the moment of inertia.
As you can see, the analysis of moments of inertia is very similar to the analysis of static moments. There are many similarities, for example:
- moments of inertia can be defined with respect to the axes of any coordinate system—regardless of where it is located or how it is oriented—though in most cases, such calculations will not have physical significance (more on this in a moment…).
- moments of inertia can be easily calculated from the above analytical formulas for all simple shapes, and then used as “ready-made formulas”—an example of a calculation for a rectangle is shown in the graphic below.
- usually, these “ready-made formulas” are associated with the centroid of the simple figure and describe the moments of inertia calculated with respect to the axes of the central coordinate system of the figure. The axes of such a system are called central axes of inertia, and the moments defined with respect to these axes are called (as you might guess…) central moments of inertia of the figure.
Now that we know what moments of inertia are from the standpoint of their definition and analytical calculations, we should move on to calculating the moments of inertia for complex sections…
But before we do that, let’s discuss the final piece of the puzzle we’ll soon be analyzing. This element is something we call the moment of deviation (product of area). Analytically, the moment of deviation is defined by a formula that, you could say, mixes what is described by static moments:
D_{xy}=∫_AxydAJust as static moments are an indicator of how evenly the area of a figure is distributed with respect to a given axis of the coordinate system (remember that when the coordinate system was centered on the centroid, the static moments were zero), the moment of deviation describes how evenly the area is distributed between the “positive” and “negative” quadrants of the coordinate system.
If we look at the elements of this integral, we’ll see the product of the x and y coordinates. This means that if a portion of the area lies in the first or third quadrant of the coordinate system, its moment of deviation will be positive (since we’re multiplying either two positive values or two negative values), whereas if a portion of the area lies in the second or fourth quadrant, its moment of deviation will be negative (we’re multiplying a positive coordinate by a negative coordinate).
In summary, the moment of deviation can be positive, negative, or zero—it all depends on how the figure is oriented relative to a given coordinate system (or how the system is oriented relative to the figure ????). To better understand this, check out the graphic below.
Let’s now move on to analyzing complex sections, which will again be similar to the analysis of static moments in its own way.
Central axes and moments of inertia – this is where it bends…
When calculating the centroid of a complex figure, we used a simplified notation for the static moment, which instead of an integral, contained a sum over simple figures.
The same applies to calculating the moments of inertia for a complex figure. It should be broken down into a set of simple shapes, and then the moments of inertia for each of them (or subtracted if a simple figure was “cut out” from the complex one) should be summed with respect to the chosen coordinate system.
However, there’s a problem! Earlier, I pointed out that
the “ready-made formulas” for moments of inertia of simple figures are for moments calculated with respect to central coordinate systems…
Does this mean that if I want to calculate the moment of inertia with respect to any axis, I have to go back to integrals…? Fortunately, no!!! Jakob Steiner comes to the rescue, making our lives easier with—guess what—the Steiner theorem:
The moment of inertia of a figure with respect to any axis is equal to the sum of the central moment of inertia relative to an axis parallel to the analyzed axis and the product of the area and the square of the distance between these axes.
In other words, if we want to calculate the moment of inertia of a simple figure with respect to a given axis, we look up the central moment in the “ready-made formulas”—with the caveat that we find it for the appropriate axis (yes… the one parallel to ours…), then find the distance between these axes (yes… the distance between the axes can only be given if they are parallel…).
Once we find the distance, we add to the central moment the product of the area and the squared distance. Either way, this formula can be expressed mathematically (e.g., for the x-axis parallel to the central axis xc):
I_x=I_{xc}+A⋅y_p^2The same applies to calculating the moment of deviation—to calculate it with respect to a given coordinate system, we sum the moment of deviation with respect to the central system of the figure and the product of the area and the coordinates of the centroid of that figure in the given system:
D_{xy}=D_{xc yc}+A⋅x_p⋅y_pIf you still have any doubts, check out the graphic below:
For reference, I’ve included the “ready-made formulas” for simple shapes ????: „gotowe wzory” dla figur prostych
Okay, to summarize…
If we want to calculate the moment of inertia of a complex figure with respect to a given axis, we:
- divide it into simple figures, calculate the central moments of those figures from the “ready-made formulas,”
- compute the total moment of inertia of the simple figures with respect to the axis using Steiner’s theorem,
- sum up the moments of the simple figures
- and voilà—we’ve calculated the moment of inertia.
Easy, right? I’ll add that sometimes it happens that the axis with respect to which we are calculating the moment of inertia passes through the centroid of one of the simple figures—in that case, for that figure, we don’t need to add the part resulting from Steiner’s theorem since the distance between the axes is zero!
Principal axes of inertia – where is it stiffer…
Now let’s move on to the topic related to the physical meaning of our calculations.
A while ago, I mentioned that the results of calculating the geometric moment of inertia would be useless if we calculated them for any random axes. Returning to where the integrals representing moments of inertia came from (the equations related to bending), I should add that the system with respect to which they should be calculated is the central system…
Hence, if the results of the calculations are to be useful for evaluating the stress or displacement of one-dimensional systems (beams or frames), we should calculate the central moments of inertia. This applies to both systems with simple cross-sections and those with complex cross-sections.
In conclusion—when performing calculations for simple cross-sections, it’s enough to use “ready-made formulas” (if the bending axis is parallel to the axis for which the formula is given!—if not, you’ll need to make a small adjustment, which I’ll discuss in a moment).
If we perform calculations for the central moments of inertia of a complex figure, we can follow a procedure that is essentially correct for all tasks:
- Divide the complex cross-section into simple shapes (the fewer, the better).
- Assume a coordinate system (I recommend placing the entire figure in the first quadrant).
- Calculate the static moments of the complex figure with respect to the chosen system.
- Calculate the coordinates of the centroid.
- Introduce a central coordinate system at the centroid (for the entire figure).
- Calculate the central moments of inertia for the simple figures (i.e., with respect to their own centroids).
- Sum the moments of the simple figures, applying Steiner’s theorem (i.e., shifting the central axes of the simple figures to the central axis of the entire figure).
- Done—you’ve calculated the central moments of inertia of the complex figure.
Of course, there will be examples where part of the calculations can be simplified (e.g., due to the presence of symmetry axes). As always, if you have any doubts—the above scheme is shown in the graphic:
Finally, I’ll explain why I mentioned the moment of deviation earlier. The matter is quite simple but requires a bit of imagination.
We already know how to define a central coordinate system and calculate the values of the central moments of inertia with respect to that system.
We also know that with respect to any system, we can calculate the moment of deviation (including with respect to the central system), which is a measure of the even distribution of the cross-sectional area relative to the coordinate system.
Now focus! Imagine I start rotating the coordinate system around its origin. During this rotation, the values of the moments of inertia and the moment of deviation relative to this system must change!
Why? Because if I rotate the system by 90°, the values of Ixc and Iyc must switch places, and the moment of deviation Dxc yc must change its sign! This means that the values of the moments of inertia depend not only on the location of the coordinate system but also on its orientation. Moreover, if the moment of deviation changes sign during this rotation, it means that for some angle of inclination of the system, it must be zero!
This angle of inclination has great significance in mechanics because it defines the principal axes of inertia, i.e., the directions of the coordinate system axes where the values of the moments of inertia are extreme! This doesn’t mean that these values perform acrobatics (sic!), but one of them is the minimum, and the other is the maximum value of the moment of inertia attainable for central coordinate systems! These extreme values are called the principal moments of inertia.
If you need help visualizing this, check out the animation:
So, in summary: for every cross-section, we can find a central coordinate system where the values of the moments of inertia are extreme. For this system, the moment of deviation is zero, meaning the area is perfectly evenly distributed between the “positive” and “negative” quadrants of this system.
A small note: in many cases, we can immediately determine the principal axes of inertia (or at least one of them)—this applies to all symmetrical cross-sections. If a section has a symmetry axis, that axis is immediately a principal axis, since the moment of deviation with respect to the system associated with that axis will always be zero.
Why are principal moments of inertia important enough to take up such a large portion of this article?
Because when analyzing bending, we talk about simple bending when it occurs relative to one of the principal central axes of inertia. For such bending, the formulas for stresses and displacements are significantly simplified (more on this in another article). Any other form of bending is considered oblique bending, where you either decompose the bending moments along the directions of the principal central axes of inertia (the formulas remain simple, but there’s more work), or you use rather complex formulas that take into account both the moments of inertia and the moment of deviation of the cross-section.
Okay—now you can calculate the moments of inertia and the moment of deviation for simple shapes, determine the central moment of inertia for a complex figure, and you know what the principal moments and axes of inertia are. In the next part of the article, let’s dive into how to analytically determine the inclination of the principal axes and the values of the principal moments of inertia.
