Analytical buckled mode shapes of ring

Hi,

I am facing problem in finding the analytical exact buckling mode shapes for ring. I have already found buckled shapes through ABAQUS, and now I am trying to find the buckled shape through some analytical equation. If anybody has some idea about it, so please let me know. Can it posible to get the exact analytical equation for buckled shape of ring which approximate match with ABAQUS result.

Anish Kumar

M.Tech. (Solid Mechanics and Design)

Mechanical engineering department

IIT Kanpur, India.

Email id:- anish [at] iitk.ac.in