As far as I can see you assumed that the middle section is perfectly rigid. With this assumption you get the same result; applying a force uniformly over a perfectly rigid element is the same as applying half of the total force on each end. Drop that assumption and you'll see a difference.
I want to make sure I understand. When you say assuming the middle section is rigid. That means the assumption that I made that the angular displacements at the two ends of the center piece are equal to zero?
I used the stiffness matrix on each element. The three elements range from 0<x<914, 914<x<2134, 2134<x<3048. I assumed since the ends were fixed that the displacements and rotations about those ends were not applicable which simplified my stiffness matrix. The nodes at my center element I assumed angular displacement was equal to zero. So my matrix looked like this.