Torsion Function Found by Separation of Variables

For certain cross sections of simple geometry, for example the circular one, closed solutions for the modified torsion function F may be obtained (14). On the other hand, for arbitrary cross sections, such as a rectangular one, it is not possible to find a solution from the boundary equation. In this case, the method of separation of variables is used, and the following solution is proposed. 

On account of the symmetry of the Y with respect to the axis OX, for all = 0 and for y = b, the following equality must be fulfilled  

These derivatives in combination with Eqs. (17.214c) and (17.214d) permit us to express the components CTj, and 3xz of the stress tensor as 

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