A vibrating column with a critical load exerted on it

[urocissa]

I was wondering the critical load vary as a result of forced vibration?
Intuitionally, I thought the critical load reduced.
But how to show it analytically, Any idea?

 

[Mapes]

Are you referring to the critical load for buckling? Why would the critical load be reduced? Once you have a mechanism in mind, you can start analyzing it.