Steel plate Bending or shear failure question

I have come across a strange situation... let me try to explain.

If one were to have a steel plate, covering the end of a pipe (1/8th thick plate, 3" id pipe) and you "pushed" a 3/8" radiused head screw into the center of the plate, how could you determine when the plate will plasticily deform, VS the the screw "punching" through the plate?

In my case, we have a new plate material that is higher in yield than what was used in the past, and now we seeing the "bolt" punch through the plate - the failure is in shear.. Ive been out of school for 20 years, and Im struggling with this. Any help would be greatly appreciated.

It may sound overly simple, but have you tried applying punching shear calculations to the situation? That should at least give you a rough approximation of when the material will cease bending and go into a shear state. If the plate is of a higher yield strength than the previous ones, it just be reaching local shear failure sooner due to the increased bending capacity.