Applying for three PhD programs at the same school I found three phd programs in one school. The programs and areas of research are close to each other and also close to my interests and I am interested in studying  in any of the programs. I contacted some faculty members and they encouraged me to apply.These programs have shared faculty members. In addition, applying for more than one program does not have additional fees.

I applied for programs a and b. I am wondering to apply for c phd program or not. Does applying for three programs have a negative effect on my admission chances? Applying for three programs means writing three SOPs to be read by a small group of professors.
This answer is coming very late, but may be useful to people who land up on this thread.

If all the three departments have faculty and research which are aligned with your research goals, there is no harm in applying to all 3 programs. The graduate committee is made of professors who are very well aware of the fact that almost all phd applicants apply to multiple phd programmes, the difference here is that they know where 2 other applications of yours went. They may ask you to set a priority, that's all. Showing interest is different from receiving an offer, professors may be interested in multiple candidates, but due to constraints give out offers to a smaller pool. They should not hold anything against you. If they do, you are better off not joining that programme.  

I applied to 3 phd programmes in the same university, and got selected in my second choice. All 3 departments knew that I had applied to the other 2 as well. They had no issues with it.