Curved beams

Abstract

The problem of the state of stress in a curved beam under load has been attacked by Winkler, and, later, by Mr. Andrews and Prof. Karl Pearson, whose paper contains an account of the prior work on the subject. Bernoulli-Eulerian methods were used in both these investigations. While the conclusions arrived at indicate that the curvature, when considerable, cannot be neglected without serious error—a matter which is borne out by experiments—the values of the stresses arrived at do not satisfy the conditions of the problem in a satisfactory manner. Now consider the effect of a pure bending moment M applied to a uniform curved narrow beam whose central plane is ABCDE and whose axis of curvature is C z , as is shown in fig. 1. The strained beam will also, by symmetry, be of uniform curvature: suppose it adjusted so that the axis of curvature C z , the plane ABCDE, and the line ABC are in their original positions. Let CA = r 1 and CB = r 2 be the inner and outer radii of the beam. If CFG be a radius of the unstrained beam in its central plane, it will strain into a radius CLN as is clear from symmetry. Thus the wedge ABGF will strain into HKNL and a point P, whose co-ordinates referred to CAB are r , θ , will strain to Q, whose co-ordinates are r + u , θ + ϕ ; u being the radial displacement and consequently ∂ u /∂ r the radial strain. The circumferential strain will be ( r + u ) ( θ + ϕ ) - r θ / rθ which is r / u + ϕ / θ and ϕ / θ since is constant everywhere, this may be written r / u + k .