ON THE DISPLACEMENTS IN THE PROBLEMSOF SAINTVENANT,AND THE CENTER OF SHEAR AND THE CENTER OF TWIST

Abstract

Consider the equilibrium of an elastic cylinder fixed at one end and loaded at the other end. The solution of this problem according to Saint-Venant is well known. This solution gives a uniquely determined stress system, but the corresponding displacement contains an arbitrary rigid body motion. In this paper, we first relax the boundary conditions at the fixed end to six conditions by an energy consideration. These conditions determine the arbitrary rigid body motion in Saint-Venant's solution uniquely. Then the translations and rotations of any transverse cross section is defined by a similar energy consideration. The center of twist is defined as the point which remains fixed during the twist of the cylinder. The center of shear is defined as such a point that when the resultant of transverse loads passes through it, transverse cross sections have no rotations about the longitudinal axis. It is shown that these two centers have identical coordinates where ψ is the warping function in the problem of torsion, and I_x ,I_y are the principal moments of: inertia of the cross section. It is proved that for constant transverse loads with parallel directions, the one which passes through the center of shear produces minimum strain energy in the cylinder.