Let’s just go to your last paragraph first. The only reason you can think of is, in fact, the basis for the disparity you’re seeing. The ‘effective length factors’ (let’s use the term “ELFs”) I’ve discussed/presented (last year, identifying 2.3 as ELF for a ladders and Xs bracing configuration, and this year identifying ‘about 0.55’ as ELF for all Xs configuration w/ 1/8” legs and 1/16x1/32 Xs) take into account – are derived from - the ‘end conditions’ at the ends of a column being evaluated/of braced segments- what they are when the legs are installed and braced. When you look into Euler’s buckling theorem a little more deeply, you’ll see that the end conditions determine the “effective length” (‘K’ in the equation you initially present). The six end condition cases typically defined/discussed include ‘fixed-fixed’ (where ends are not free to either rotate or translate), “pinned-pinned” (where the ends are free to rotate, but not translate), and four other combinations.
It is important to understand that end conditions and their associated effective lengths are not…. a limited finite set, but rather, they are a continuum – actual end conditions can be… anywhere in between; for instance, not fully prevented from translating or rotating. So, a real end condition can be somewhere in between, and calculating the effective length can get really complex and intimidating.
When you do a SFPD BS measurement of a stick, the end conditions of that stick are pretty close to (they approach being) “pinned/pinned” (both ends are able to rotate – but not ‘translate, i.e., move side to side) as load reaches critical load and the stick starts to bow/buckle). If you were to test that same stick under “fixed-fixed” end conditions (when the ends are not able to either rotate or translate), you would see a measured BS about 2.3 x what you saw under “pinned-pinned” end conditions. When you look at the ends of leg segments braced with ladders and Xs, the end conditions of the segments approach “fixed-fixed” end conditions. The way the ELFs are used are to adjust SFPD (pinned-pinned end conditions) BS measurements to the end conditions of braced segments in the installed legs. It is the adjusted stick BS you do inverse square calcs on to determine the braced interval length needed to brace the leg to your design load.
I did a couple long posts last year explaining how and why an ELF of about 2.3 for ladders and Xs is consistent with the theorem and, it turns out, predictive in use: See this Wikipedia page for discussion of effective length and end conditions -
https://en.wikipedia.org/wiki/Euler%27s_critical_load
Here's one of those posts:
Getting into buckling a little more deeply:
Now there’s one more twist/complication in translating what you measure on the scale, doing single finger push-down testing, to the buckling strength you’ll get when the tested piece is in-place, and braced, in the tower. The length term in Euler’s equation is actually expressed as “effective length.” This term depends on the “end conditions” of the column/piece in question I’ve explained this in detail in the thread “Measuring/using Buckling Strength- New Information”.
Briefly, when you do the single finger push-down test, you’re applying/creating “pinned/pinned” end conditions. At each end of the stick being tested, the end is able to rotate- as the stick buckles, and the part near the end deflects from vertical, the end, instead of staying horizontal…tips/rotates. Now when its in-place in the tower, and braced, looking at each of the braced points, the end of each braced segment is not free to rotate; its held in place; the ends of each braced segment is firmly attached to, is a part of the adjacent braced segment. This end condition is described as “fixed-fixed.”
The effective length of a column in pinned-pinned end conditions is 1; the effective length of a column in fixed-fixed end conditions is 0.65. (Note, this is the “recommended design value for K”- see drawing on linked Wikipedia page) Going back to our general equation BS = 1/(Proportion (of L2 to L1) squared), we calculate 1/0.65 squared, and we get 2.37 . What that’s saying is if a given stick is tested under pinned-pinned end conditions, if it were tested under fixed-fixed end conditions, the buckling strength measured will be about 2.3 times what it measured under pinned-pinned end conditions. You’d see something close to this result if you did single finger test, then glued the bottom of the stick to the scale pan, and glued the top to a plate (so it was perpendicular to the plate), and pushed the top plate in a way it stayed parallel to the scale pan.
To that discussion, let me add: fixed-fixed end conditions allow neither rotation or translation- the ends can’t move laterally- side to side, nor can they rotate (like they can doing SFPD testing)- in the example of glueing the ends to scale plate and to a plate at the push-down end, and similarly in the installed tower leg, where the ends of two adjacent braced segments are effectively glued to each other- they’re just the same stick continuing through the braced point, and the bracing fully prevents movement/translation. The buckling strength of the ladders prevents …almost any movement of the braced point(s) toward adjacent legs, and, acting independently, the tensile strength of the Xs prevents almost any movement of the braced point(s) away from adjacent legs
As I’ve posted before, the effective length factor of ‘about 0.55’ for all Xs was a simple back calculation from data on two different State winning “all Xs” towers. The specific data we had to work with on one tower was tower did not meet 29cm circle bonus, so leg forces at full load were 3777gr; tower was braced at 1/11 interval, held full load; legs (1/8”) had 36” stick SFPD measured at “between 25 and 26gr”. At this bracing interval (5.62cm), proportion of braced interval to 36” (91.6cm) was 0.061, so 1/proportion squared was 265.83. So, the question was, what was the in-place BS, which when braced at 5.62cm intervals w/ all Xs, got to a braced strength of 3770gr? Turned out to be 14.196gr. 26 divided by 14.196 = 0.546.
As I’ve discussed/said a number of times, I do not ‘have a solid theoretical handle’ on the basis for this factor like I do for a ladders and Xs bracing configuration. It is clear from… many people’s designs/experience, that for a given leg strength, you have to use a tighter bracing interval with all Xs than you do with ladders and Xs bracing- the question is how much tighter, which comes down to what ELF to apply. The braced point(s) in all Xs bracing is effectively prevented from moving away from adjacent legs by the tensile strength of the Xs, just like in ladders and Xs; no difference in those planes/directions. But the buckling strength of the Xs in all Xs, acting to prevent inward bowing/movement - translation of braced point(s) toward adjacent legs is a lot less than what is available with ladders and Xs- there is less to prevent translation in those directions. When you look at the linked Wikipedia page – the end conditions and corresponding effective lengths, looking at the case of rotation fixed & translation free, you see an effective length of 2.0. Applying that effective length would mean a calculated ELF of 0.25 (1/2^2). Just as with ladders and Xs, rotation is fixed/prevented by the fact that the end of one braced segment is fixed to the end of the adjacent segment. However, there is some….greater degree of freedom for translation (of adjacent legs toward each other) than in a ladders and Xs bracing configuration. Interestingly, if you use an effective length of 1.33 (instead of 2), you get an ELF (1/1.33^2) of 0.557. I suspect, given the continuum nature of end conditions, that may be what’s going on, but I have no idea of the factors one would have to know and mathematically take into account to derive that value.
I don't see any obvious arithmetic errors in your analysis. Regarding a few numbers in your analysis:
Wonder what the SFPD BS of those 1.6gr/36” sticks was…did you measure?? On E values, a while back I posted a link to an old US Forest Service study of E vs density for balsa. Their data set covered densities of 0.08 to 0.19 specific gravity, and showed the value of E in this range going from about 200,000 to 800,000 psi (which translates to 1379 Mpa tp 5515 Mpa)