8Plate Torsion

In this section, calculations are made step by step, from the simplest to the most complicated, due to the complexity of the description of deformations.

8.1 Torsion of homogeneous cylinder

Consider a cylinder of length L along the torsion axis x. It has a radius R along y and z and a torsion test is applied, imposing a rotation by an angle θ; since there is no warping of the transverse section, the longitudinal displacement is zero. The two displacements v and w due to the rotation by an angle θ imposed by the torsion are v=-zθ and w=yθ. The following shears are generated:

[8.1]

[8.2]

In order to write the Lagrangian, an integration over the whole transverse surface of the infinitesimal volume element ds=2πrdr must be made with relation r²=y²+z². Denoting by F_T the torsion resonance frequency and by G the shear modulus, the deformation and displacement energies are:

[8.3]