The stability of a ring under the action of a linear torque, constant along the perimeter

Exact analytical solutions of problems on the static and dynamic forms of the loss of stability of a ring, under the action of a linear torque constant along the perimeter, are found using the consistent equations of the theory of plane curvilinear rods constructed earlier taking account of transverse shears. Two forms of torsion of the ring are examined: the external forces creating a torque remain in the plane of a cross-section of the ring in its initial undeformed state (“dead” forces, case 1) or in its deformed state (“follower” forces, case 2). It is shown that, in the second case, the solution of the static instability problem found is practically identical to the solution of the problem corresponding to the dynamic formulation and is reduced to an examination of the oscillations about the static equilibrium position. In the case of both forms of loading, loss of stability of the ring occurs without deformation of its axial line, with it bending predominantly in the plane of the ring accompanied by a slight distortion. It is established that a study of the forms of loss of stability of the ring for the type of loading considered is only possible using the equations constructed, taking account of transverse shear.