Well, I eventually did make a calculation of the MEW-published tripod according to Euler's critical load formula for buckling columns. I'll copy-paste in the whole calculation below, along with a short discussion.
The executive summary is this: I used a spreadsheet, and ensured that the spreadsheet would replicate another example I found on the web — i.e. the spreadsheet does not contain any silly mistakes. I used the dimensions mentioned in the MEW article. I de-rated the Euler's calculation result by a safety factor of 2. The resulting maximum allowable load — by Euler's theory, not by field test — was 4200kg. In contrast, a similarly-sized commercial steel tripod (by Spanco) is rated at 1800kg for the heavy-duty version – with legs that look a lot chunkier than the 48mm scaffold poles here. So I would suggest that the simple Euler's calculation is not the full story.
For those that might like to see the nuts and bolts, here's my working-out and fuller discussion:
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Tripod rated load – theoretical calculation by Euler’s critical load formula
As the legs on a tripod are slender columns, assume critical stress is lower than yield stress, and assume the failure mode will be buckling, where the column is subjected to a load greater than the tensile strength along the column’s long axis. Load is assumed to be purely axial without any eccentric load, so no bending stresses are calculated. This also implies that the load must not be swung in case it hits one of the legs. It is assumed that the legs are restrained from movement at their base by a lashing rope and at their head by pinned connections to a cap. The load is a suspended between the legs by an eyebolt centrally placed in the cap.
The calculation below relies on finding the critical load leading to failure, using Euler’s formula for critical load. This formula provides a value for the ultimate load, but to be sure that the structure will not fail, an ultimate factor of safety has to be applied, so that the structure will be sure to support the rated load without failing even if there are local imperfections in the materials.
Material: Standard steel scaffold poles
Modulus of Elasticity, E (approx.) = 200 GPa
Outside dia, Do = 48.3 mm
Inside dia, Di = 40.3 mm
Length of legs, L = 3.05 m
Angle of legs to ground = 70 degrees
Ultimate factor of safety (failure load/allowable load), FS = 2
Axial load capacity of one leg:
Area moment of inertia, I = (Do^4 – Di^4)*pi/64
where Do and Di are expressed in metres
I = (0.0483^4-0.0403^4)*pi/64 = 0.000000138 = 1.38*10^-7
Euler’s critical load:
Pcr = pi^2 * E * I / (K * L)^2
where
Pcr = Euler's critical load (longitudinal compression load on column),
E = modulus of elasticity of column material (200 GPa, expressed as 200*10^9 Pa),
I = minimum area moment of inertia of the cross section of the column,
L = unsupported length of column,
K = column effective length factor (take K as 1.0, i.e. both ends pinned)
Pcr = pi^2 * 200*10^9 * 1.37*10-7 / (1 * 3.05)^2 = 29214 N
Calculate the rated axial load capacity, Pax, of one leg:
(Pax is the force directed along the inclined leg)
Pax = Pcr / FS
Pax = axial load capacity of one leg = 29214 / 2 = 14607 N
Calculate Pz, the vertical component of Pax:
Pz = Pax * sin (70) = 13726 N
Calculate the total load for the three legs:
Total load Pztot = 3 * Pz = 41178 N
To obtain maximum permissible load in kilogram:
convert value for force in Newton to kilogram-force, where 1 kgf = 9.807 N
Pztot / 9.807 = 4200 kg
The result: according to the Euler buckling analysis, maximum permissible load on the tripod is 4200kg
This would need to be field tested by suspending a load 125% of rated capacity, i.e. 5250 kg to prove that 4200 kg is indeed the permissible maximum load rating.
The maximum load value changes as the legs are splayed more or less. If legs are splayed further out to a 60 degree angle, the maximum load decreases from 4200 kg to 3870 kg; whereas if the legs are moved in to an 80 degree angle the maximum load increases to 4400 kg.
In reality, commercial tripods in this size have far more conservative maximum load ratings. Spanco, an American company, sells steel tripods with legs that can adjust from 3.09m in length to 4.93m. They make the tripods in two grades, one rated to 1 ton (900kg) maximum load, and the other to 2 ton (1800kg). Incidentally, the Spanco tripod product manual states that the legs should not be splayed out to more than what works out to be about 70 degrees angle.
The comparison of the Euler's analysis to the commercial practice casts doubt on the usefulness of Euler's analysis, at least on its own, as a way to assess a load rating for a hoisting tripod.
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