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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.

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To have an itroduction you can have a look at this book where it is explained what you are asking (if I have understood correctly the question). In chapter 1 you can see an example of axially loaded rod solved with the PVW and in parallel discretized through the FEM and the solutions are compared. Later it continues with problems with a bigger dimensionality and with different discretizations in 2D. For something similar I suggest you also the text by Quarteroni (for sure more based on the numerics) where there is a clear example of how the FEM is used in structural mechanics (and you will find also an extension to fluid mechanics). In this book you can also find the DGSEM method, where the FEM is "mixed" with Finite Volume.

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Broadly speaking the PDE in structural engineering is the PDE of elasticty or elastoplasticity. This can be small or large deformation depending on whether you are solving static problems or problems in which large deformation is expected, such as shock and blast. For elements such as beams, shells, plates, rods you can derive specialized pdes starting from the general equations of elasticity.

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