The main reason you're not getting a smooth shade on your mesh is because one quarter of the surface has normals facing the opposite direction. You can view this by going to Display > Custom Polygon Display Options Box and turning on Face Normals. You'll notice that the black shading strips are along the edges where faces have extreme opposite normal directions.

Here is a simple way to fix this:

1. Select the mesh.
2. Conform the normals to face the same direction (Edit Polygons > Normals > Conform).
3. Reset the normals to get rid of the black strips (Edit Polygons > Normals > Set to Face).
4. Average the normals for a smooth shade (Edit Polygons > Normals > Average Normals).

Hope this helps.