I only just started thinking about this, so I apologize if I can't frame it correctly... But say I have a completely uniform beam sitting on, say, 8 supports all distributed evenly about the beam's center of mass (which is also its geometric center). That is, for every one of the 4 supports on the left side of the center of mass, there is an equivalent support the same distance from the center of mass on the right side.
Let's say I know the mass of the beam and the distances from the center of mass of the 8 supports. It seems like the symmetry of this problem makes finding the force exerted by the beam on any one support impossible. That is, we can only find information about the distribution of forces... eg, all the supports on the left add up to be taking only 1/2 the total weight of the beam, etc.
I'd like to ask if this is true. Even though I know the mass of the beam, that the beam is uniform, and the distances of each support from the center of mass of the beam, is ultimately the force on each support from the beam arbitrary - IE, the only constraint is the way the forces add up? If I (in the "real world") take a more or less uniform beam, arrange supports more or less symmetrically around its center of mass, and find the force on each support, how will the distribution look? Will all the forces on the left side add up to the same thing as all the forces on the right?
More specifically. Say my 8 supports are symmetric about the CoM. Say the beam is 50 ft long. The first support on the left and the first support on the right are 5 ft away from the CoM. The second support on the left and the second support on the right are 15 ft away from the CoM. The third support on the left and the third support on the right are 18 ft away from the CoM. The fourth support on the left and the fourth support on the right are 25 ft away from the CoM, IE at the ends of the beam.
In this scenario, the supports are still symmetric about the CoM. But the distribution on each side is still uneven, IE the first support is 5 ft away, the second 15 ft, the third 18 ft, the fourth 25 ft, corresponding to spacings between supports of 10, 3 and 7 ft.
Is finding the forces on the supports still arbitrary?
Sigma F = 0
Sigma M = 0
The Attempt at a Solution
I wrote out the equations of equilibrium for the beam and support system. I saw no way of pulling out of just these equations and the information I know (mass of beam, distances from CoM of supports) the forces on each beam.
Is this how it is? Or am I overlooking something? If the supports were not symmetric about the CoM, could I find the force on each?
How would an engineer do this problem? If I have an actual beam that will rest on actual supports distributed in some way under the beam, how can I predict the force on each support (not taking into consideration bending or flexing of the beam, lack of uniformity, etc.)?
If the supports were on a circular track, the beam attached to the supports (no sliding), and the support/beam system were rotating at some rate, would the distribution of forces on the supports change? (I think it should!) In this case could I calculate the forces on individual supports?
Thank you so much... this is driving me crazy.