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How Our Craft Stick Beams Resist Bending.

Four 2 X 3 Popsicle Columns

Cross section view of possible craft stick beams.

The problem we have with the craft sticks and bending is bending in the thin direction (axis). The answer I have come up with is stacking the sticks to get a wider beam in that direction as shown in the first picture.  This post will show why by using the principles discussed in the last post, “How a beam resists bending. (Moment of Inertia)” .

In the next picture I show a cross section view of 4 possible craft stick beams.   So far I am constructing the main beams using the configuration of the two center cross sections, 2 or 3 sticks wide.   However, we will calculate one 4 sticks wide and a single stick.  The unit I will use is 1 = stick thickness and the area is 1 for a single stick in cross section.
I have color coded each stick according to its position from the center of the beam.  The center of the beam is marked by the circle with the cross in it.   The direction of the bending we are interested in is the vertical direction on the picture.

Since the purpose of this is just to get a feel for things we are going to do a lot of approximations and be a little loose with the math.  I am going to take an average distance from each stick to the centroid.  This is sloppy math, but good enough to show some interesting results.  I will talk more about how it should be done at the end of this post.

The formula for the moment of Inertia is I=AXdXd where A is the area of the section and d is the distance from the centroid.  (dXd = d squared, this software does not have the ability so do superscripts.).  The easiest example to start with is the 2 sticks in the  2nd picture (the red sticks).  The distance from the centroid to the center of the sticks is 1 and 1X1 = 1.  The area, A, of each stick is 1 but there are two sticks so A= 2.   This gives us an I of 2 (in our strange units.)

Now lets do the same calculations for the green sticks in 3 stick beam.  The center of the green sticks is 2 from the centroid.  So d=2 and dXd =4  again our total area A = 2  and this gives an I of 8 for the green sticks.

Next we will do the same calculations for the blue sticks in the 4 stick beam.  For these d = 3 so dXd = 9 and 9X2=18 for I for the blue sticks.    We can finish the calculations for the total I for the beam because we already know the I for the red sticks.   The total I for the 4 stick beam is 18 + 2 = 20.

Now let’s go back to the magenta stick for the single stick and the center of the 3 stick beam.  If we imagine splitting the stick in the middle at the centroid, then the average distance for each 1/2 is 1/4.   1/4 * 1/4 = 1/16 and the total area A is 1.  So for a single stick we have an I of 1/16.  No wonder it is easy to bend a single stick.

Resistance to bending of the 4 beams

The results of these calculations is shown in the third picture.   As I stated earlier we were playing a little loose with the calculations but the results are not all that bad.  Using calculus the real real relative resistance to bending comes out to be:

1/12 for magenta
2 1/6 for red
8 1/4 for green
18 1/6 for blue

The reason our method was a little loose with the math is we assumed the average was the center of each beam, or 1/2 beam in the magenta case.  Using the magenta example, the bottom part of the top one half of the beam cannot supply any resistance because it is exactly on center.   The top edge of that 1/2 of the beam is doing most of the work so it is obvious that the average is not in the center of that one half of the beam.  However, our results were good enough to understand what was going on.

If a computer program was used to do these calculations, it would take small horizontal slices of the beam and calculate the results exactly as we did.   Then it would take even smaller slices and compare the results.  The computer would continue taking smaller and smaller slices until the difference between two consecutive calculations was very small and then report the results as “good enough”.

An even greater source of error in our calculations is a hidden assumption.  It is assumed when doing these calculations that all parts of the beam are fixed to each other.  Our craft sticks are only fixed to each other at the ends of each stick.  This means that as the sticks are bending, the spacing between the sticks is probably not what we are using in the calculations.  However, this gives us a really good idea of what is going on and the reason I chose to build the main columns of the tower in the 3 stick wide and 2 stick wide configuration.

In the next post about these sticks we will talk about a final thing to consider called radius of gyration.  This number and the total length of a column will help determine if a column will buckle under a compressive load.

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