TABLE OF CONTENTS Structural members, such as arches, crane hooks, chain links, and frames of some machines, that have considerable initial curvature in the plane of loading are called curved beams. The flexure formula of Art. 5.5.10, f = Mc/ I, cannot be applied to them with any reasonable degree of accuracy unless the depth of the beam is small compared with the radius of curvature.
Unlike the condition in straight beams, unit strains in curved beams are not proportional to the distance from the neutral surface, and the centroidal axis does not coincide with the neutral axis. Hence the stress distribution on a section is not linear but more like the distribution shown in Fig. 5.42c.
Just as for straight beams, the assumption that plane sections before bending remain plane after bending generally holds for curved beams. So the total strains are proportional to the distance from the neutral axis. But since the fibers are initially of unequal length, the unit strains are a more complex function of this distance. In Fig. 5.42a, for example, the bending couples have rotated section AB of the curved beam into section A ?B ? through an angle ? d ?. If ? o is the unit strain at the centroidal axis and ? is the angular unit strain ? d ?/ d ?, then the unit strain...
