Torsion Of Circular Bar Research Articles

Torsion of cylindrically orthotropic elastic circular bars with radial inhomogeneity: some exact solutions and end effects

Torsion of elastic circular bars of radially inhomogeneous, cylindrically orthotropic materials is studied with emphasis on the end effects. To examine the conjecture of Saint-Venant’s torsion, we consider torsion of circular bars with one end fixed and the other end free on which tractions that results in a pure torque are prescribed arbitrarily over the free end surface. Exact solutions that satisfy the prescribed boundary conditions point by point over the entire boundary surfaces are derived in a unified manner for cylindrically orthotropic bars with or without radial inhomogeneity and for their counterparts of Saint-Venant’s torsion. Stress diffusion due to the end effect is examined in the light of the exact solutions.

Analysis for Beam/Bar Structures Based on Homogenization Method.

A homogenization method based on the finite element method for beam/bar structures subject to torsion, bending and axial stretch/compression, is derived. The detailed configurations of the beam/bar structures are assumed to be periodic. They are composed of repeating unit cells. The unit cell is represented by three dimensional finite element model. Displacement fields based on the three dimensional finite element model for the unit cell and those based on Euler-Bernoulli hypotheses for the bending deformation of beam and also based on the torsion of circular bar, are superposed, representing a complete deformation field within the structures. Proposed formulation is, in rigorously speaking, valid when the size of the cross section of the beam/bar structure approaches zero in limiting sense. However, as an approximation, the theory can be applied to the case that the cross section of the beam/bar structure is finite but is small compared with their length. Some numerical examples are presented to discuss the validity of the proposed method. The proposed homogenization technique can be applied to analyses for composite beam/bar and also for other types of structures having complex internal structures, such as holes of complex shape, complex cross sectional shape, etc.