Consider a circular beam of constant rectangular cross-section bent by end moments in the plane of curvature.

**Pure bending of
a Beam of Circular Arc **

Consider a circular
beam of constant rectangular cross-section bent by end moments in the plane of
curvature (Figure 6.5). Since the moment diagram is con-stant, stresses are
only a function of *r* and a stress function of type I is called for:

Equations (6.24) and
(6.25) can now be evaluated for the displacements since all the constants *A*,
*B*, *C*, *D*, *F*, *H*, and *K* are known.

The rotation and the
displacements for a particular beam can be computed from the Winkler and
elasticity solutions for comparison. Results for a deep beam (Figure 6.6) are
presented in Table 6.2. The Winkler or elementary val-ues of the displacements
at the free end, Î¸ = Ï€/2, are computed by
integration

Comparing the two
solutions, we see that again, agreement is excellent. At the neutral axis, *u _{r}*
is the same. The elasticity values are computed assuming

Although the radial
displacement at the centerline depends slightly on Pois-sonâ€™s ratio due to the *K*
term, the small radial change in thickness:

Thus, considering both
stresses and displacements, the close agreement of values computed from the
simple Winkler analysis indicates that the added refinement from elasticity
theory, while very satisfying, is of little practical significance.

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Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : Pure bending of a Beam of Circular Arc |

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