Materials for shaft

Start with the basics:

• What is the shaft for? Gear box, helicopter, hydroelectric turbine, ............?
• What, if anything do you want to optimize? Weight, corrosion resistance, cost, size/strength?
• What are the environmental conditions? Underwater, corrosive (what type), high temperature, .........?
• What type of loading? Steady, reversing, shock loaded, bending, .........?
  

As guest already told

start with boundary conditions

- Housing - bearing - Id = shaft OD

- Coupling etc

- design the shaft dimensions

- calculate the weakest point as per stress is concerned.

- Select the material having sufficient factor of safety corresponding to this zone.

The factor of safety will depend upon the type of loading.

- refer to any machine design book, Shaft design is the very basic in it.

When in doubt make it stout out of something you know all about!