Simple Beam Problem: Solving for Vertical Forces and Symmetry in Mechanics

[Demian^^]

I'm trying to refresh my mechanics by solving some fairly simple, self invented, beam problems. This goes fine, but I can't figure out how to solve the following one.. it sounds simple enough, though..
Imagine a (two dimensional) beam of size d with a vertical outside distributed constant downward force acting on it (for instance gravity).
It is supported by four points that can exert a vertical force on the beam, to keep it in balance. They are placed completely symmetrical: two of them at the outsides of the beam and two at a distance, for instance, d/10 away from the center of the beam. Their forces can be named Va,Vb,Vc,Vd or something. When i use the symmetric properties I can reduce the number of unknowns to two, but then I'm stuck. I can use Sum_verticals=0 and Sum_Moments=0, but they both give the same relation between the two unknowns that are left over.
Is there a fundamental reason why this cannot be solved this way? Does the outcome mean that there is in fact one degree of freedom left? This does not sound very physical, as I can imagine that the forces exerted by the points on the beam will always be the same in a true situation.
Could anyone help me on this one? I'm probably making some very stupid mistakes.
Thanks!

 

[Doc Al]

Think about it: Imagine four people carrying a huge log, distributed as you have stated. Now, without changing any of the positions of the vertical supports, one of the guys in the middle could ease up on his lifting (thus increasing the load on the others) or he could lift harder (decreasing the load on the others). Can you see why there isn't enough information to solve this?

 

[Demian^^]

Yes, I'd figured that out, more or less. I wonder, however, if I could add extra equations to solve the problem. If my intuition is right and all the supporting points are equal, then it sounds logical that everytime you would repeat the experiment, they would exert the same reaction force and not suddenly the two middlepoints each 5N more and the outer ones 5N less. So I assume the problem as stated here is too 'simplified' and there are other factors that play a role?
(Thanks for the reply!)

 

[Pyrrhus]

Well the system you described is an Hyperstatic system of degree one. It can't be solved only by the equations of static. You need also forces-displacement relations (constitutive equations and compatibility equations).

Check https://www.physicsforums.com/showthread.php?t=80873" for more information.