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Membrane theory applies to shells and is a different thing entirely. Membrane analogy applies to open or closed sections. If you have a copy of 'Design of Welded Steel Structures' by Blodgett, look at page 2.10-25 (Fig. 45) to see the shape of the membrane for various shapes including a channel and a WF.I think the membrane theory applies to closed sections.
That is an inaccuracy. Torsion causes a force in the top and bottom flange in opposite directions. The forces are not lateral. They are parallel to the flange in the rotated position. Together, they form a couple.Torsion causes a lateral force in the top and bottom fla in opposite directions.
This is not a valid procedure. The direction of the flange force is constantly changing throughout the beam span.Case 1..no stiffeners
Check the fla as a bm with this load on it. It will have supports at each end. Inbetween it will have a continous spring support whose spring stiffeness is equal to the bending stiffeness of the web in double curvature. Find max deflection.
Again, not a valid procedure. The stiffeners are attached to a beam which has rotated but the cross section is otherwise unchanged, so stiffeners have no effect.Case 2...add transverse stiff.
Same model as case 1 but add conc spring supports at each stiffener location whose spring stiffeness is equal to the strong-axis bending stiffeness of the stiffeners in double curvature. Find max deflection.