Bending of large curvature beams.: I. Stress method approach

Abstract

The paper presents a coherent theory of the uniform bending problem in a circular curved beam, with multi-connected cross-section, having a large radius of curvature with respect to its width. The three-dimensional elastic problem is solved, in the case of linear homogeneous isotropic body, assuming the stress tensor as the unknown and by exactly satisfying the field compatibility equations. The mathematical structure of the governing boundary value problem (BVP), enlightened here for the first time, is unexpectedly complicated: a fourth-order elliptic (variable coefficients) partial differential equation with two degenerate unstable boundary conditions (i.e. involving second and third order partial derivatives in a direction that becomes tangent at several points of the boundary). Such a kind of BVP seems to be typical of the curved beam bending problem since it also appears in the displacement approach ( Mentrasti, 2001. part II, Int. J. Solids Struct. 38, 5727–5745). As a final point, it is reduced to a simpler problem by an ad hoc integral representation, assuming the ρ-convexity of the cross-section domain.