The book I'm referring to says that a shaft with a circular cross-section in pure torsion will have its cross-sections flat during the loading. The cross sections won't deform; they will rotate.
It then says that this is not the case for non-circular cross-sections. In a non-circular c/s shaft, the c/s distort and are not flat during the loading.
I wanted to know why that is so.
We say and write what we observe. In history, when an axisymmetric cross-sectional beam (i.e. the beam's cross-section still remains the same when rotated about its longitudinal centroidal axis) was subjected to torsion, the cross-sections of the beam remained plane/flat. This is true for both, solid beam and hollow beam. However, when a non-axisymmetric cross-sectional beam (i.e. the beam's cross-section DOESN'T remain the same when rotated about its longitudinal centroidal axis) was subjected to torsion, it was observed that the cross-sections didn't remain plane/flat. In fact, they warped when subjected to twisting. The general equation for calculating stresses due to pure torsion for a solid circular beam is shown below:
where T: Torque applied, r: distance from the centroid, J: Polar moment of inertia. Now, the derivation of this equation involves an assumption that the plane section must remain plane, i.e. the cross-section cannot warp. Therefore, this equation cannot be applied to any other non-axisymmetric cross-sectional beams.
Warping basically refers to one-half of the cross section (above/below the neutral axis) being subjected to compression, and the other half to tension. This behavior is what we usually see in a beam subjected to bending, however, this behavior is also observed in non-axisymmetric cross-sections subjected to torsion only. Calculating shear stresses for a non-circular cross-section is somewhat more complex and complicated than the circular ones.
not so much a formal answer (more like an answer for the intuition), is that during torsion on non circular shafts, the stresses tend to deform the material in such a way that the cross-section resembles more that of a circular.
In order for that to happen , material needs to be deformed/pulled from adjacent section. The end result is that it contracts.
A non-circular section under the torsion deforms in both St Venant shear and also by warping.
For example, an I beam deforms both by warping and shear deformation. Warping creates tension and compression in flanges. so the plane of the flange does not remain flat anymore.
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