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For torsional constant, analytical formulas are available for some standard steel sections as shown in this wiki page.

But what about arbitrary shapes? How can I obtain the formulas for torsional constants? If no such formulas are available, are there any first principles that I can use to deduce the approximate formulas or FEM calculation for torsional constant?

Note: a related question about the FEM formulation of torsional constant is being asked here.

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For arbitrary thick-walled shapes, there are no generic formulas available. You have to calculate the torsion constant using numerical methods, e.g. FEM or BEM.

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    $\begingroup$ Do you have any pointers on how to derive FEM solution for torsional constant ? $\endgroup$ – Graviton Dec 22 '15 at 13:01
  • $\begingroup$ Sorry, I don't. But I just found this document that seems to explain how to calculate the torsional constant numerically: people.virginia.edu/~ttb/torsion.pdf $\endgroup$ – Robin Dec 22 '15 at 13:46
  • $\begingroup$ Do you have any pointers on how to derive FEM solution for torsional constant ? – Graviton. This seems like a really, really good stand-alone question. I'd recommend you ask that on its own. $\endgroup$ – Mark Dec 22 '15 at 19:28
  • $\begingroup$ @Mark, I've asked my question here $\endgroup$ – Graviton Dec 22 '15 at 23:54

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