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I'm asked to analytically calculate the composite torsional constant of the following structure.

enter image description here

No other information is given, such as the applied torque.

I know how to calculate the torsional constant for each component (an equilateral triangle and a semicircle), but I don't think I can just add them weighted to the areas.

Is there anything else I can try?

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You need to find the shear flows in each individual cell, and then apply the usual torsion constant formula to the cells and add the result.

You can get one equation from the equilibrium of the "corner points" where the sections join (there are two points, at the ends of the vertical rib, but by symmetry you get the same equation for each point).

To get another equation, you can to assume each cell has the same amount of twist - i.e. the cross-section does not warp under torsion.

This works through the details: http://web.aeromech.usyd.edu.au//structures/acs1-p83.html. Google "torsion multicell beams" for other references.

If you want to include warping of the cross section, the best way to deal with this type of structure is model the correct geometry in a finite element analysis, rather than trying to reduce it to a "beam element" Aside from the assumptions made in calculating a single torsion constant, the actual torsional stiffness is very sensitive to the way the ends of the "beam" are connected to the rest of the structure, which may or may not prevent warping or other deformations of the cross section.

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