Towers B/C

[cool hand luke]

if you can find a machine shop with a water jet you can get the jig Balsa Man described cut out. from 16 guage steel.

[Balsa Man]

if you can find a machine shop with a water jet you can get the jig Balsa Man described cut out. from 16 guage steel.

Yeah, and welded together. Man, that would be quite a weapon

[Gr8tor]

Thanks for the detailed post about jigs, Balsa Man. I appreciate it. It would be helpful if you could supplement that with a drawing or picture

[cool hand luke]

if you can find a machine shop with a water jet you can get the jig Balsa Man described cut out. from 16 guage steel.

Yeah, and welded together. Man, that would be quite a weapon

don't say that

I'll probably get in trouble now for sending a weapon to school with the kids.

[jsiongsin]

How do you calculate the density needed for a certain part of the tower, such as the leg? I have already read the past forums, but I may have missed some posts detailing how to do this.

[Random Human]

How do you calculate the density needed for a certain part of the tower, such as the leg? I have already read the past forums, but I may have missed some posts detailing how to do this.

Density needed?
Density is mass/volume

I'll give you some numbers, 36*1/8*1/8 = .5625 thats your volume
36*1/16*1/16 =
0.140625

then just divide your mass by those... not sure what you mean really...

[Balsa Man]

How do you calculate the density needed for a certain part of the tower, such as the leg? I have already read the past forums, but I may have missed some posts detailing how to do this.

Not sure I understand, either, but I think I’m hearing “how do you calculate the density needed for a part to be strong enough to hold?.”

If that’s the question, the answer is, there’s not any way to calculate the density needed.

What there is a way to do, though, is … get to a good estimate of the target density range, for buying/ordering/selecting wood. Then it’s a matter of sorting through your wood to find the ‘best’ sticks. I’ve discussed this process at some length, in a number of past posts; by asking this question the way you have, it sounds like you’ve indeed missed something fundamental, or aren’t yet fully grasping some important, basic concepts, and how to use them. So its worth going over these things again.

It is not density value(s) you need to design a tower that will carry a design/full load, it is strength values, to make sure the pieces you use are strong enough to carry the forces they will see.

You can measure the strength of a long (like 36”) stick; you can calculate its strength when its braced to shorter sections, and you can measure stick weight/density. What you want to be able to do is to find (from the” lumber pile” you have to work with) the lightest sticks that have your “design strength.” Doing this is a statistical and budget problem/challenge.

Strength is related to density; as density goes up, so does strength, whether its buckling strength, or tensile strength. For the rest of this discussion, I’m just going to deal with buckling strength. It is buckling strength (along with the braced interval and configuration of bracing system) that determines if a leg, or a set of legs will work (as in carry the design load). Buckling strength (BS) depends on the length, cross section (technically cross sectional moment of inertia, aka 2nd moment of inertia), and inherent stiffness (“E”- modulus of elasticity, aka Young’s Modulus). In a leg, braced into a set of shorter (much stronger) ‘stacked columns’, BS also depends on the bracing interval (which defines the length of the shorter ‘stacked columns’), and the bracing system used (e.g., “ladders and Xs”, or “all Xs). This is all from Euler’s Buckling Theorem. Euler’s Theorem assumes a) a ‘long thin column’, b) an elastic material, and c) a homogeneous material. Because of the nature of wood, which we’ll get into in a minute, “calculations” using these factors are approximations.

With an “inverse square table” (I posted a link earlier to the one we used for last year’s towers), you can calculate the in-place BS, given a measured ‘single finger push-down’ (SFPD) BS at 36”, at whatever bracing interval you’re looking at, and depending on the bracing system/configuration being used. BS is proportional to 1 over the length squared (our inverse square relationship); cut a stick in half, the BS of the halves will have about 4 times the BS of the full length stick; into thirds, 9 times, and so on. Actually, going one level deeper into the math of Euler’s Theorem, this inverse square relationship is to the “effective length.” Effective length takes into account the ‘end conditions’ of a column being evaluated.

When you measure the BS of a stick put vertically on a scale, pushing down on the top of the stick with one finger placed on top (“single finger push-down”), you are creating/using end conditions that approach “pinned-pinned”; as the stick bows/deflects/buckles under axial loading, the stick is able to rotate about the ends. Were you to do this test with the end on the scale glued to the scale plate, and the upper end glued to a horizontal plate that is pushed down (staying horizontal), you would be testing under “fixed-fixed” end conditions (stick can’t rotate about the ends).

When you brace the legs using a “ladders and Xs” bracing configuration (ladders glued in between the legs, with X strips running between the points where the ladders are), you have created end conditions that closely approach fixed-fixed end conditions. The effective length factor for this situation is about 2.3. What that means in calculations is you measure BS at, say 36” (which = 91.6cm), and its 40gr. You multiply that measurement by 2.3 (which = 92gr). Let’s say we’re evaluating a chimney section leg segment (40cm leg segment length, at a 1/6 bracing interval). 1/6 of 40cm is 6.666cm; your bracing interval points along the leg segment are 6.666cm apart, so the proportion that 6.666 is of the 91.6cm length tested at is 0.07278. 1 over the proportion (0.07278) squared is 188.7876. This is telling us that each braced segment (and hence the entire leg segment made up of braced/stacked segments) has a strength of 92 times 188.7876, which =17368gr. Assuming vertical chimney legs, the force at full tower load (15000gr) is ¼ of the full load, which = 3750gr. So a 40cm leg segment from a stick at that measured BS, braced to a 1/6 bracing interval will have a strength that is over 4 times the strength needed.

The effective length factor for an “all Xs” bracing configuration (using 1/16” x 1/32” for the X strips is, as I’ve posted discussion of (back on page 1 of this thread), very different from the 2.3 for ladders and Xs; it is about 0.55. So, again, with a 36” SFPD BS of 40; 40 x 0.55= 22gr. Using 22gr instead of 90gr in the calculation above, we end up with an in-place BS of 4153.327gr. That’s only a bit over the 3750gr force on the leg segment; it gives you a bit over a 10% safety factor.

So, we’ve gone over the things you can measure and calculate; we know how to approximately calculate the strength needed, and what in-place strength a braced leg at a measured BS (and measured stick weight/density) will give you.

On average, if you double the density (2x), strength goes up by about 2.25x). The ‘on average’ statement is important, because wood is grown, not made. Measuring the strength (BS) of two sticks weighing the same, you will likely see a difference. If you cut a stick in half, the BSs of the halves will likely be different from each other, and from the original stick. This “variability around the mean” is a very important concept to understand.

This means, for an example, if we look at a bunch of 1.4gr/36” sticks, we’ll see most with a 36” SFPD BS around 37-38gr. With a big enough sample/wood pile, you’ll see some/a few with a BS up into the low 40s, and a very few as high as 45, 47, maybe 47. Conversely, some will be as weak as 30gr, maybe even a bit less. There’s 20% or more plus or minus variation. So, if your design calculations and inverse square calcs say you need a 36” BS at, say 40gr, braced w/ all Xs to carry your design force, a) if you order a bunch of 1.4gr/36” sticks, you have a decent chance of getting some, a reasonable number that meet the needed strength. Some will not be strong enough. If you order some 1.3gr sticks, a few, maybe a very few of them will have a BS as high as 40gr. The key to being able to… intelligently get/order/buy wood is data that shows what the variability around the mean of density vs BS looks like.

This graph plots the density (expressed as stick weight at 36” length) of 1/8” sticks vs their BS (measured ‘single finger push down’ testing at 36”. There are over 400 sticks in the database this plot is made from. That’s a large enough “n” (number of samples) to get a decent statistical handle on the situation.
Link
https://docs.google.com/document/d/1I0Z ... sp=sharing

In this graph, you’ll see the data points, and the linear trend line (the dashed blue line). That trendline runs through essentially the middle of the data points. If you look along the bottom/x-axis to a given stick weight (say 1.4gr), and go up to the trend line and go over to the vertical y-axis, you’ll see a corresponding BS value of a bit under 40gr. In a pile of 1.4gr sticks, many/most will have this BS. If you add/draw a line, running above the trend line, and essentially catching/tracking the best/highest BS values seen at given densities/stick weights, it will show you the highest BS you are likely going to be able to find/see at a given stick weight; a very few sticks in a large pile of 1.4gr sticks will show a BS at or close to this “best” value.

Putting this data; this understanding, this picture of the relationship of BS to density (stick weight) together with an inverse square table is how we can “get to” the answer to the question we started with- in what density range of sticks am I likely to find a given BS, and what are the lightest sticks that may have the strength I’ll need, at a given bracing interval. This is what such an inverse square table looks like for all Xs (at 1/16 x 1/32).

Link

https://docs.google.com/spreadsheets/d/ ... sp=sharing

Enjoy….. these two tools, together give you a basic 'engineering manual' for tower design. You can now do set up a spreadsheet to look, quantitatively at the key design tradeoff you want to figure out - at what combination of leg strength (and approximate weight) and bracing interval do you get the lightest tower? Lower density/'floppier' legs with a tighter bracing interval, or higher density/stiffer/stronger legs with a wider bracing interval; what combination optimizes this tradeoff?

[Balsa Man]

Thanks for the detailed post about jigs, Balsa Man. I appreciate it. It would be helpful if you could supplement that with a drawing or picture

You're welcome. Looking for something like this?

https://docs.google.com/document/d/1Fql ... sp=sharing

[retired1]

Definitely a candidate for laser cutting.

[Gr8tor]

Thanks Balsa Man. You won't believe how much that picture helped me. I have one question. The legs of the tower won't be flat against the jig. Right? Will that be a problem?