In uniform torsion about the -axis, the axial displacement describes departure of a cross-section from a plane. It is determined up to an additive axial translation by , , where is the torsion stress function for a simply connected bar. For a circular section these derivatives vanish. For an ellipse they give , so a noncircular section generally warps. The stress-function Poisson equation is exactly the mixed-derivative compatibility condition for this axial displacement.
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