I have learned about the finite element method (FEM) as a method for solving boundary problems given by a PDE. The way I learned it is to approximate the solution by a linear combination of test functions. (I think it is called "Method of mean weighted residuals".)
Now when looking for FEM online I see it is used in structural engineering (and that that was the field of study where FEM was developed).
In these structural engineering texts, there is no mention of a PDE that is numerically solved. (See, for example, this lecture)
Is the FEM as used in structural engineering the same as (or equivalent to) what I learned, and if so which PDE is solved in the problem of structural engineering.
With FEM as used in structural engineering I mean the calculation of the displacements of the nodes of the elements.