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Parallel chord space frame - Equivalent beam

Parallel chord space frame - Equivalent beam

Parallel chord space frame - Equivalent beam

(OP)
I came across old documentation regarding the topic of the subject and I need to evaluate the validity of the methodology.

The frame is represented as follows;
https://files.engineering.com/getfile.aspx?folder=...

Although the figure is 2D, the frame is a 3D frame.

To reduce the frame into a equivalent beam, the documentation provides formulas for the beam section properties. While the Bending quantities are trivial (derived from Steiner's rule), I'm struggling with the torsion rigidity. The formulas for the torsion rigidity are

https://files.engineering.com/getfile.aspx?folder=...

The quantities of the formulas are related to first figure and A_1 and A_2 are the cross-section areas of the diagonal members on perpendicular sides of the frame.


My interpretion of the formula is that the 1/2 * B^2 * A_1 and 1/2 * H ^2 * A_2 has to do with Steiner's rule and the Sines and Cosines are transformations (which I cant figure out).

Can someone explain (in detail) how one arrives to the formulas for I_vy and I_vz ? Or are the formulas erroneous ?

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