Bending of large curvature beams.: II. Displacement method approach

Abstract

A large curvature circular beam subject to uniform bending in its plane is investigated by a displacement method. This is the companion of a previous paper in which the same subject has been investigated by a stress method approach (Part I, Int. J. Solids Struct. 38 (2001) 5703–5726). The functional form of the three-dimensional displacement field is determined exactly: the cross-section remains plane and rotates about the Z-axis, while each longitudinal fiber remains circular. This result is independent of the constitutive equation of the material, provided it is compatible with the uniform bending hypothesis. The equilibrium equation (in several form) and the boundary conditions are derived for the linear elastic and homogeneous beam: they constitute an unstable degenerate boundary value problem with a structure similar to that found in Part I. The non-variational nature of this kind of problem is also detected. Finally, the relationship with the potential stress function ψ is derived, governed by a hyperbolic second order partial differential equation (another unusual occurrence appearing in bending problem).