You absolutely can do FEA by hand. Finite element analysis (FEA) is just math. In the pre-digital age, performance calculations were done on paper using a physical prototype during testing.
The difference today is that the algorithms driving simulation software can perform FEA calculations much faster and can test for multiple force effects simultaneously.
It also means you don’t need to build a physical model to apply calculations—the system can apply them to your 3D design without leaving your desktop.
While FEA by hand is still feasible for simple geometries or when a prototype already exists, FEA simulation software can test the performance of ever-more complicated geometries, so the design process isn’t constrained by what you can measure by hand.