How to Determine Vertical Loads on Bearings in a Statically Indeterminate Shaft?

[Spimon]

Homework Statement

Determine the load (vertical only) on the three bearings at A, B and C. The shaft is loaded at one end with force F (see attached diagram).

Homework Equations

Cantilever Beam Deflection
Deflection at End = PL^3/(3EI)
Deflection at Point X = Px^2/(6EI)*(3l-x)

The Attempt at a Solution

I'm having a few difficulties remembering the details of solving statically indeterminate problems. My attempt is as follows:

1. Remove reaction forces B and C. Applying beam deflection formulas I can find the displacement of points B and C due to force F. I called these δB1 and δC1.

2. Replace the reaction forces at B and C, but remove the force F. I can now find the displacement of points B and C due to the reaction forces, respectively. I called these δB2 and δC2.

3. Since the actual deflection at the fixed points B and C is zero, I can set:

δB1 + δB2 = 0

and similarly

δC1 = δC2

This leaves 2 equations and 2 unknowns which may be solved.Any hints, comments or corrections would be a world of help

 

[PhanthomJay]

All supports take vertical load only (simple supports) and cannot carry moment, so the support at A is not fixed and cannot act that way when supports B and C are removed. The beam is statically indeterminate to the first degree, so you need to initially remove just one support... like perhaps B.