CURVED BEAMS

The flexure formula applies to a straight member, because the normal
strain within the member varies linearly from the neutral axis. If the
member is curved, however, the strain will not be linear, and so we must
develop another method to describe the stress distribution. In this section
we will consider the analysis of a curved beam, that is, a member that has
a curved axis and is subjected to bending. Typical examples include hooks
and rings. In all cases, the members are not slender, but rather have a
sharp curve, and their cross-sectional dimensions will be large compared
with their radius of curvature.
The following analysis assumes that the cross section is constant and
has an axis of symmetry that is perpendicular to the direction of the
applied moment M, Fig. 6–40a. This moment is positive if it tends to
straighten out the member. Also, the material is homogeneous and
isotropic, and it behaves in a linear elastic manner when the load is
applied. Like the case of a straight beam, we will also assume that the
cross sections of the member remain plane after the moment is applied.
Furthermore, any distortion of the cross section within its own plane, as
caused by Poisson’s effect, will be neglected.
To perform the analysis, three radii, extending from the center of
curvature O of the member, are identified in Fig. 6–40a. Here r references
the known location of the centroid for the cross-sectional area, R
references the yet unspecified location of the neutral axis, and r locates
the arbitrary point or area element dA on the cross section