Suppose you have a beam with a transverse load P pointing down in the center, directly over the middle "spring". This beam is supported by 9 uniformly distributed springs of constant k. Compute the loads of the springs.
You know that the problem is symmetric, and that the ends of the beam go off to infinity. You, thus, do not know the displacement of the endpoints. You know the spacing of the springs. This beam is assumed to have no mass.
The Attempt at a Solution
Basically, I have been racking my brain over how to set up the system of equations.
Since the problem is symmetric, I know that there are only 5 unknowns (spring loads 1 2 3 4 = springs 9 8 7 6, respectively)
I know that, since the beam is symmetric, the slope in the middle is 0. If I deconstruct the beam into halves, I can combine the loads of 4 of the springs to an initial shear force where the load is being applied, as well as an initial moment.
So, equation 1: slope in the middle is 0.
I need 4 more equations, but I do not know where to retrieve them from. In this case, there are no points where the slope is 0, or where the displacement is 0. I cannot find any points where the shear is 0, or where the moment is 0.
How do I find further boundary conditions to solve this?