Equilibrium Equations for Torsion

The substitution of the components of the stress tensor, given in the preceding section, into the equilibrium equations [Eq. (4.14) with = 0] leads to 

Taking derivatives in the first two equations with respect to x and y, respectively, and adding the resulting expressions, we obtain 

Disregarding the trivial case, / = 0, Eqs. (17.216a) and (17.215c) imply that 

As in the elastic case, it can be demonstrated (1, p. 63) that if the torsion is produced by the application of a torque at one end of the rod, whereas the other end is fixed, then on the basis of the minimum for the total energy of the distorted bar, the fibers parallel to the torsion axis will be distorted along 

To determine the boundary conditions on the surface of the rod, it should be noted that for thin rods the external forces acting on the lateral surface are considered negligible in comparison with the internal tensions, and the following equation holds for the sides of the rod  

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