In summary, the person completed both problems and got the same answer for both. They were wondering if this was possible and if the two point loads were equivalent to the distributed load. The stiffness matrix was used on each element, with the assumption that the middle section is perfectly rigid. This assumption simplifies the stiffness matrix and results in the same answer. They also mention using three beam elements and assuming fixed ends and zero angular displacement at the center element.
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.