I recommend that you get a copy of the following paper "Reinforcing Loaded Steel Compression Member:, AISC Engineering Journal, fourth quarter 1988. You can obtain this from the AISC website.
The existing dead load and residual stress in the core section may have an effect. The existing dead load can cause the core section to reach its buckling load before the reinforcement. When this happens a plastic hinge forms in the core, and the reinforcement must provide all stability. This behavior is different from beams, where the stresses from the unreinforced and reinforced sections can be superimposed if traditional ASD is used.
The paper has criteria that determine when you can design the reinforced column and ignore the existing load, and when the existing load must be taken into consideration. You calculate 2 lengths for a given reinforced column, L1 and L2. If the unbraced length is less than L1, the full properties of the reinforced column can be used. If the unbraced length is greater than L2, the column should be designed for elastic behavior only. If the unbraced length is between L1 and L2, the existing column core will fail first, and the strength is based on the post-buckling strength provided by the reinforcement.