lllazar wrote:Well, if you want a rectangular base, im assuming youd do something like 20 cm x W cm, w/e W may be? The fact is, with a base height of 15, the legs would be under a lot of stress due to the angle at which they are from vertical - a square base in this sense would be more practical.
But don't take my word for it, im sure other more experienced builders will try to answer your question to the best of their abilities.
Illazar, you’re absolutely right that the more the legs are leaning, the more force they have to carry. Per earlier post, if legs were vertical, they would carry ¼ (for a 4-legged tower), or 1/3 (for a 3-legged tower) of 15kg. As the angle from vertical increases, the load increases as a function of 1 over the cosine of the angle. The trick is understanding what “the angle” is. You want to minimize that angle to minimize the force the legs need to handle, because the less the force, the lighter the legs can be. What's a little tricky here is visualizing/understanding things in three dimensions.
At the base/bottom, the legs need to span the 20cm square hole in the test base. At the 15 cm (30cm for B), the legs need to fit inside an 8cm circle. For a square base, by putting the legs at the mid-points of the sides of the 20x20 hole, you end up with the legs (on each of the sides), about 16cm apart. At the top, the distance apart is the side of a square that will fit in an 8cm circle- I don’t have drawings with me, but let’s say for discussion purposes, that distance is 6cm. The base portion looks like a truncated pyramid – a square w/16 cm sides at the bottom; a square with 6cm sides at the top. The legs lean in toward each other, and in toward the center of the tower.
OK, now let’s consider a rectangular base. Looking at one side, the legs, at the base, need to be 20+ cm apart (tapering in to end up 6cm apart at the top. Presumably, these sides would be put together 5cm apart - Illazar's "W" (so the legs at the top of the tower will fit under the 5cm square load block. You’d have two flat sides, and two sides where the base is leaned in quite a bit, and the upper portion is leaned in slightly. The lower portion would be…..very much like a short bridge; clear span of 20cm, top spam of 6cm, width of 5cm.
For simplicity, let’s just consider the base portion, now. It would seem, at first thought that the legs in the rectangular configuration would be angled in more; 20 cm apart (as opposed to 16cm) at the bottom, and 6cm at the top. However, let’s look more closely at the square. If you look at the plane defined by two opposite sides, the base ends of the legs are 20+ cm apart, and the top ends fit inside the 8cm circle (our assumed 6cm apart); this is the same angle as in the rectangular case. It is the angle in toward the centerline of the tower. If we look at the base “pyramid” from one side- looking at the…..outline, or silhouette, or shadow, we see legs leaning in from 20+ cm to 6cm, BUT, those legs are in a tilted plane – a plane tilted from the base in toward the center of the tower. The angle of that tilt is the same angle as we see in looking at the outline. Looked at in this view, It’s a compound angle. However, by rotating the tower/base section we’re looking at 90 degrees – rotating around the centerline of the tower, now the outline we’re looking at has a pair of legs at the outside (a pair that are diagonally apart from each other), and the other pair of legs line up with each other- the front leg lined up on top of the back leg. The angle the outside legs lean in toward each other – and, this is the important part; in the plane we’re now looking at, toward the centerline of the tower, is the same as the angle they lean in at in one of our flat sides from the rectangular configuration. So, “the” angle; the angle that determines how much more force the legs see than if they were vertical, is the same in both cases.
So which is “more efficient”? (as in lighter). If you figure a ladder piece joining the tops, and a ladder piece joining the mid-points of the legs, in the rectangular, you have a total of 60cm, and in the square 68cm – a little over 10% in favor of the rectangle. but, the ladder piece in two of the sides is longer (13 cm vs 11cm)- meaning they have to be stiffer (i.e., heavier), while on two of the sides they are a lot shorter (5cm vs 11cm), i.e., a lot lighter. You’ll have to figure out how these trade-offs work.
There is, I believe a big downside to the rectangular configuration (with flat, parallel sides), though; overall tower stability. With the load on top of the tower 50cm above the base, only a tiny amount of bucket sway across (perpendicular to) the plane of the flat sides will create a force that will act to tip the tower over. Likewise, only a tiny amount of distortion out of the plane(s) of the flat sides will do the same. With all sides leaning in some toward the center you gain overall stability against both distortion and bucket sway.
I think a – perhaps the - really important question this year is how do the tradeoffs between a square (4-legged) tower, and triangular (3-legged) tower work out……….