projecting triangulated mesh edges on original mesh

Hello!

I'm trying to get that the edges of a triangulated mesh and project them back to the original mesh, and to give them volume (height and width)

I tried to use weaver bird bevel edges, then thicken and then cutting them with my original mesh and and offset mesh in the height I want, but it doesn't work

I need them separate from each other so I can later remove them manually. 

What would be the best way to do this?

Thanks