Weak formulations (a.k.a. variational formulations) of partial differential equations (PDEs) are hugely important in the FEM as they enable the concepts of linear algebra in the analysis of PDEs. This concept transform PDEs into sets of linear equations (a matrix) that can eventually be manipulated and inverted using standard matrix methods.
Transforming an equation from strong to weak form requires the use of virtual function, and hence the name Principle of Virtual Work.
Also read
A few worked examples (external pdf)
The role of the Principle of Virtual Work in FEM
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