fanjiatian wrote:How can you figure out an appropriate geometry for bracing between compression and tension members without relentlessly testing the milllions of possibilities?
By understanding some basics; what the forces at work are, and what do to to the pieces/members in the boom; by doing a bit of studying/thinking, and getting to understanding how buckling works, and then applying that understanding. There has been some explanation and discussion this year in the Boom threads (in both this and the Scores thread). As noted in those discussions, there was a lot of discussion/explanation of this last year in the Tower thread, and going back further, also discussed & explained in Elevated Bridge threads.
The purpose and (possible-depending on your design) need for such bracing is
solely for preventing buckling/increasing the
buckling strength of the compression members.
As a point of clarification, retired1's comment, "Using 3/32 bass wood for the tension members requires zero trussing."; (cross-sectional) size (and wood)
does not matter. The tension members require no bracing; there are
no forces at work trying to move/bend them out of alignment. The load they see is pure, "axial" tension- one end is being pulled away from the other; that pull/tension force is along the line- the long axis - of the tension member. The higher that force gets, the more in a straight line the member is pulled. If the rules allowed it, a kevlar string would work fine (kevlar because it stretches so little under load).
There's no off-axis force to be braced against, or that needs to be controlled against/prevented.
The compression members are a entirely different critter. Assuming a "good" alignment - straight/symmetrical/precise, the the force on them is an axial compression force; the ends are being pushed together-
toward each other. As that force increases, at some point, the member "buckles"- it starts to bow; to bend in the middle. If the force continues, the bowing out continues, and the member experiences "buckling failure"- it breaks.. As I've suggested before- go to Wikipedia; "buckling." There you will see Euler's Buckling Theorem, and his equation that explains/quantifies how it works, for long, thin columns- which is what we have in a boom compression member.
While you're studying this, to give yourself a feel of how it works, get yourself a small stick of balsa- say a piece of 1/16th x 1/16th. Cut a piece, say a foot long. Put one end on a scale- standing vertically. Push down on the top. Note the weight (force) on the scale when it buckles (starts to bow in the middle). Now take a piece (from the same stick) 6" long, and do the thing. What you will see (with the accuracy depending on how close to vertical you do the alignment/pushing, and how close the the density (inherent strength) of the two pieces are, is that the force at which buckling happens- the buckling strength - has gone up by a factor of 4. If the foot long piece buckled at 10gr, the piece half as long will buckle at 40gr. If you go to a piece 3" long (1/2 the 6" piece), the buckling strength will go to 160gr. This is called an inverse square relationship-
the buckling strength has an inverse square relationship to the "effective length" of the column. You'll see this in Euler's equation.
When you cut the length in half, the buckling strength goes up by 1 over 2 squared- 4.
What bracing does is change
the effective length of the compression member. If you (effectively/correctly) brace the midpoint, you create two "stacked columns" with a buckling strength 4x that of the unbraced one. Just like in towers, with bracing between the 4 legs. The spacing of this bracing should be equal, so all the braced sections (the effective column length) is the same. If one section is longer, that is where buckling will happen.
So this means there aren't "millions" of efficient options, just a few "integral" ones- bracing at 1/2, 1/3, 1/4th, 1/5th, etc.
The 4cm spacing retired1 suggests would be bracing at 1/10 of the 40cm length. ,A piece braced at 1/8th intervals will have 100 times (10 squared) the buckling strength of the full, unbraced length. With two compression members (in a C- division boom), they each need to carry about 20kg (about 15kg in a B-div boom). If, as it sounds like, he's saying, they're running 1/8th inch square-24 lb/cu ft balsa, braced at 1/10th length interval (i.e., 4cm spacing), that suggests the buckling strength of a 40cm long piece at 0.2kg (200gr). That actually seems quite a bit low for that size/density- maybe worth checking with a full length/scale buckling test.
So what does "correct" bracing entail? Doing the push a stick down on a scale exercise, which direction did the buckling happen in? Toward one of the faces- not toward one of the diagonal corners. This is because the cross section (distance) between two opposite faces is shorter/smaller than the cross section/distance between two opposite corners. This is what the "I" term in Euler's equation is all about- the cross section- technically the cross sectional moment of inertia; buckling is going to happen in the plane(s) with the shortest/lowest cross section. For a square compression member, that's toward the faces, so to brace against that, you need bracing on/against two adjacent faces- at 90 degrees- just like in a tower- ladders between the 4 legs.
So, let's look at things end-on- from the distal end toward the wall- two opposite sides of the compression are vertically aligned- in the "y-plane", and the other two horizontally aligned- in the "x-plane." Bracing in the x-plane is cake- pieces between the two compression members. They can be, as retired1 suggests, very light. As buckling starts, the lateral/bowing force is
very low. All the brace has to do is resist that first bit of force- to hold things in place. The y-plane is more problematic. If you put a y-plane brace on the top face of a compression member, the only place you have to put the other end of that brace is against the tension member. In most boom layouts with 2 tension members and two compression members, the tension member is not vertically aligned with the compression member- they are not both in a vertically aligned "y-plane"; the tension member is running at an angle to the compression member; the plane they're in (again, looking toward the wall), at an angle- in neither the "x" nor "y" planes. Now we know that buckling (for a square cross section compression member) is going to happen toward one of the faces- in the x or y plane. If the brace on the top side of the compression member is at an angle, what happens as the compression member starts to buckle upward in the y plane?
It's not just putting an axial compression load on the brace, it's also putting a significant bending force on it. That means it needs to be strong (i.e., heavy) enough to resist that bending force- not just strong enough to resist buckling under axial load.
The additional strength/weight to do that is....significant, especially in longer pieces as you get toward the wall....
What all this suggests to me is that an approach that doesn't need y-plane (the vertical plane) buckling bracing (i.e., bracing between tension and compression members is the more efficient way to go. A way you can do this is compression members with a rectangular cross section- where the vertical dimension is longer, and it is that vertical dimension/cross section that provides the stiffness to resist buckling in the vertical/y-plane. The horizontal dimension/cross section can be pretty small, because you can easily/effeciently brace between the compression members in the x-/horizontal plane. You can further increase the stiffness in the vertical plane by laminating two pieces together- say 2 pieces 1/16th thick, by maybe 1/2" wide- you'd have to play with the vertical dimension- I'm just throwing the 1/2" out as a guess/for discussion purposes. The glue plane significantly increases stiffness in the vertical plane because it's modulus of elasticity ("E" in Euler's equation) is a lot higher that that of balsa. There are also other options to get the stiffness needed in the compression members- box-beam construction being one of them. A single, constructed compression member
is a viable way of getting the job done (efficiently), too.
So, food for thought and experimentation......
