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In this question I ask about how to perform a first-principle derivation of the torsional constant of a section. It appears that there is no such analytical derivation for torsional constant, so my question therefore becomes: what about FEM derivation for torsional constant of arbitrary shapes section?

Note that I am not interesting in just using a FEM package without actually understanding the basic principle. I want to be able to derive the FEM formulation from first principles.

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You can find an implementation of a finite element used in computation of arbitrary shape section torsional constant here:

http://projects.ce.berkeley.edu/feap/elmt11.f

I don't see any reference in the implementation other than a comment in the source code: "St. Venant torsion problem solution by warping and Prandtl stress function."

I don't see any reference on their website either, but hope the source would give you an idea. http://www.ce.berkeley.edu/projects/feap/

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  • $\begingroup$ The first link is now dead $\endgroup$
    – Graviton
    May 25, 2018 at 2:50
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This is a problem which is usually solved in books on elasticity theory. The underlying math is based on the solution to the Laplace PDE.

If you do a Google search for Larry J. Segerlind's book "Applied Finite Element Analysis Second Edition", he lays out the math behind calculating J using the Prandtl stress function method (see Chap. 8 'Torsion of Non-circular Sections'). There is another method which is derived using a warping function, and the math is similar. The derivation using either method requires some sophisticated math, so be prepared.

If you can't find the Segerlind book, there are others on programming the finite element method which use this same problem as an illustration. It's also used in the more advanced Boundary Element Method as well.

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Make a long shaft with your chosen section. Fix one end by 3dof restraints at each node. Put a closing plate across the other end and apply a torque, hence back calculate your torsional constant. Double the length of the shaft and check you get the same answer. St Venant's principle is the reason you want a long shaft to do this test.

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