The third article of the MechaDevs Blog will cover a very simple (and really schematic) topic: the calculation of cross-section parameters for beams, frames, and trusses.
Before reading this article, it’s definitely essential to first check out the article on internal force analysis in beams. In that article, I presented a significant introduction to why cross-section analysis is necessary and often becomes the main stage in designing various types of structures.
This article is divided into two parts. In this part, we will discuss the absolute basics, including:
- Why do we calculate cross-section parameters?
- Static moments and the centroid of the cross-section
- Examples of simple cross-sections
Why do we calculate cross-section parameters – the importance of the cross-section
JIf you haven’t read the article on internal forces despite my recommendation, I’ll briefly summarize the essence of what was written there in the context of calculating cross-section parameters.
I would say that the most important concepts in this whole topic are stresses and displacements. Both of these, in the case of calculating one-dimensional objects (those where the length is much greater than the other dimensions), are directly dependent on support, loads, and, of course, the cross-section parameters. The previous article focused on how the support method and applied loads affect the strength (specifically, the internal force diagrams).
Here, as the title suggests, we will discuss which cross-section parameters are important in the context of stress and displacement analysis in a beam.
As a reminder,
the cross-section of a beam is a flat figure obtained by cutting the beam with a plane perpendicular to its axis
Such a geometric figure has a set of characteristics that fully define it and affect the strength of the entire beam (in terms of the stresses present within it) and its stiffness (i.e., its resistance to displacement under load). Some of these crucial characteristics are well known to you, even from elementary school, for example, the cross-sectional area (A) and the location of the centroid (cx, cy).
Why do I claim that such simple-to-determine cross-section characteristics are so important? Because, in the case of pure tension or compression of a beam, it is the cross-sectional area that affects the state of stresses and displacements.
The remaining characteristics are significant in other strength cases. For example, if we analyze the stress state during simple bending, it will be necessary to determine another characteristic: the central moment of inertia (second moment of area) of the cross-section (let’s assume that bending occurs around an axis that is the principal central axis of inertia—but more on that later ????).
For shear stress analysis during shearing, we will also use the static moments (for the cut part of the cross-section—more on this in the article on stresses). In the case of torsion, we analyze the torsional stiffness factor, whose calculations are more complicated—I will devote an additional article to that.
To summarize the above considerations:
when analyzing displacements and stresses in beams, frames, and trusses, various parameters related to the cross-section geometry of the element are used
Now, let’s move on to determining the centroid of the cross-section, whose location is very important in bending system calculations.
Static moments and the centroid of the cross-section – area, area, where’s the area…
Before we proceed with any calculations, it’s worth considering a few things related to the shapes of cross-sections commonly used in industry.
Besides the fact that very often sections based on basic geometric shapes (e.g., circular and square sections) are used, which we will call simple sections, the vast majority of typical sections (e.g., circular tubes, square tubes, tees, angles, etc.) can be represented as combinations of several simple shapes (sometimes applying certain simplifications—e.g., removing small roundings, etc.). In such cases, we talk about the analysis of complex sections.
As it will soon become apparent, complex section analysis mainly involves dividing the section into a set of simple shapes and determining certain parameters for each simple shape individually. This is convenient because, for simple shapes, certain geometric parameters can be very easily determined analytically, so in further calculations, we can use “ready-made formulas.”
But what was I supposed to write about…?
Ah yes, calculating the centroid… Well, let’s start big
the centroid of a cross-section is the point where, if we place a coordinate system, the static moments of this cross-section with respect to the inserted axes will be 0
Hmm, that sounded enigmatic, especially since a term immediately appeared here that hasn’t been discussed anywhere before—specifically, the static moment (first moment of area) of the cross-section. Let’s quickly explain what a static moment is and how it’s calculated in typical problems…
If we wanted to complicate this topic, we could refer to theory and mathematics. Then we would say that static moments are defined for the axes of a given coordinate system and a given figure using the formulas:
S_x=∫_Ay dA,
S_y=∫_Ax dA
where: x and y are the coordinates in a given system, and dA is the smallest element of the area of this figure.
I know how much you love integrals (especially over an area), but believe me, for simple shapes, these can be calculated very easily. Using these formulas and the definition of the centroid, it is thus easy to determine (parametrically) the locations of the centroids for figures such as rectangles, right triangles, isosceles triangles, circular segments, etc.
As I mentioned above, these parametrically calculated locations for simple shapes are our “ready-made formulas,” which we can use for calculations of complex shapes.
So, what should we remember when calculating static moments for complex figures—this will be very simple:
- firstly – to remind you, the static moment of a figure is determined with respect to some axis of some coordinate system.
- secondly – the static moment of a simple figure with respect to a given axis is equal to the product of the area of that figure and the distance from the axis to the centroid of the simple figure.
- thirdly – the distance between the axis and the point is always measured perpendicular to that axis.
- lastly – if a complex figure is created by adding simple sections, the static moments of these simple sections are also added. If a simple figure is “cut out” from another figure or a set of simple figures, its static moment is subtracted.
Using the above points and knowing the centroid locations for simple shapes, we can simplify the calculations of the static moment of a complex figure with respect to a given coordinate system. This simplification involves replacing the dreaded integral with a simple sum:
S_x=\displaystyle\sum_{i=1}^nc_{yi}⋅A_i,S_y=\displaystyle\sum_{i=1}^nc_{xi}⋅A_iwhere cx and cy are the coordinates of the centroids of the simple figure in a given system and Ai is its area.
We perform this sum for all the simple figures from which the complex section was created. If you have any doubts, check out the animation below:
Two more small notes—always take a moment to think about dividing the complex section into simple shapes!
Following the spirit of laziness (but also optimization!), the fewer simple sections, the fewer calculations, and therefore the less room for calculation errors (which, believe me, happen quite often!).
Secondly, it can sometimes happen that when calculating static moments, you get negative results. This is not a problem because the result depends on the location of the assumed coordinate system—it means that the centroid point has a negative coordinate. For the static moment to always be positive, the system should be introduced so that the entire figure is located in the first quadrant of the coordinate system—then all the centroid coordinates of the simple shapes are positive.
I think I can now assume that you are able to calculate the static moment of any complex section with respect to any coordinate system. So, let’s move on to calculating the location of the centroid—after all, that’s what this all started with… In reality, knowing the static moment values of a given figure with respect to the axes of a given coordinate system, calculating the centroid location is a mere formality. You just need to use the formulas:
C=(c_x,c_y ) → c_x=\frac{S_y}{A}=\frac{∑c_{xi}A_i }{A},c_y=\frac{S_y}{A}=\frac{∑c_{yi}A_i }{A}As you can see, the coordinates of the centroid in the adopted coordinate system (the one for which we calculated the static moments) are simply the quotient of the static moments with respect to the appropriate axes and the total area of the figure.
If we expanded these formulas (by inserting the simplified definition of the static moment), it would turn out that the centroid coordinates result from the weighted average of the centroids of the constituent figures, where the areas of those figures are the weights ????.
Note: these calculations can sometimes be simplified (or actually skipped…) when dealing with a symmetrical figure. If the figure has a symmetry axis, the centroid will definitely lie on that axis. If it has more than one axis, the centroid will definitely lie at their intersection!
Once we calculate the location of the centroid of the entire complex figure, we can confidently introduce a new coordinate system at that point. Such a coordinate system will be called the central coordinate system.
If we were bored or had too much time, we could recalculate the static moments with respect to the central coordinate system… What do you think—what values of S_xc and S_yc should we get?
Yes! According to the definition of the centroid given above, the static moments with respect to the central axes are zero!
Now, to wrap things up, we can express this definition a bit more clearly
the centroid is the point with respect to which the cross-sectional area is evenly distributed.
Examples of simple cross-sections – back to preschool…
The last topic discussed in this article will be the issue of simple cross-sections.
After all, we’ve been assuming all along that we already know everything—the formulas for areas, the positions of centroids, etc… Well… practice suggests that sometimes it’s better to review because it’s always possible to forget something.
For this reason, I’ve presented below the basic parameters of simple shapes that most often appear in centroid analyses of complex figures.
Additionally, the graphic will also include formulas for the central moments of inertia and deviation, which will be explained and used in the second part of the article.
Alright—you now know what the centroid of a cross-section is, what its location depends on, and why static moments are needed to determine it. Now it would be good to find out what the geometric moments of inertia of the cross-section are—check out the second part of the article!
