How to cut an object

Hello, i have a problem which is how to rotate a cube and not to move the base of it.
So i thought that i can set up the cube in its half coordinates (it seems that rotate-handle is in the middle of the cube) and than i can just rotate without moving it. But i need to cut the half which shouldnt be moved. (kind of up-down plane moving an elbow)
So the problem is :
How to rotate without moving the base of primitive, or how to cut a part of an object ( i used glClipPlane but it cuts a half of whole render area).
Big thanks for any ideas.

You will need to Split your model in to 2 more more seperate parts for drawing such that you can

Draw Part A

Draw Part B

Draw Part C

Once you split your models drawing you can manipulate the seperate parts as needed…