I'm trying to solve this statics problem to find the internal tension that results, in each member:
It shows 3 idealised weightless rigid members hinged in a vertical plane, in an isoceles triangle. Two of the members carry identical loads at their midpoint (W = Mg). The bottom member is constrained in such a way that the internal tension exactly balances the "spreading" forces developed by the two members it supports.
But when I try and calculate it (which should be simple!) I get an indeterminate solution. it looks like there should be another constraint, beyond the usual ones, but I can't see any constraint that I'm missing.
As a thought experiment, it should have a single, static, unique, solution. If I built a rigid triangle like this, from 3 lengths of wood, dangled weights from 2 of them, and somehow balanced it vertically with one bottom corner on a brick and the other bottom corner level with it on a roller skate (or just stood it up in an ice-rink), that would match the setup. The resulting concoction would clearly have a static equilibrium (ignoring any bending or overbalancing!) and therefore a unique + static set of tensions, for any given reasonable lengths of wood and magnitudes of weight.
My attempt at solving
By symmetry, I should be able to analyse just one of the diagonal members, to solve for all tensions. So I just draw one diagonal strut, and the reactions/forces at its ends:
For simplicity I'm using resolved reaction forces F1 and F2 for the support at the apex from the other diagonal member, and F2 and F3 for the horizontal internal tension and roller support at the bottom right corner (the horizontal components must be equal and opposite):
2W = F1 + F3
=> F3 = 2W - F1
- Horizontally: zero net force (only 2 forces, equal and opposite)
- Moments round top-left end:
F3.(2d) = (2W).d + F2.(2h)
=> F3d = Wd + F2h
Substituting from (1):
( 2W - F1 ).d = Wd + F2h
=> Wd = F1d + F2h
- Moments round bottom-right end:
(2W).d = F1.(2d) + F2.(2h)
=> Wd = F1d + F2h
Same equation as (3).
So I now have 2 equations in 3 unknowns. I'm guessing there should be an extra constraint, but I can't see it.