As for straight beams, it is possible to develop a strength-of-materials-type solution for curved beams from the assumption that plane sections remain plain during flexure.

**Simplified
Analysis of Curved Beams **

As for straight beams, it is possible to develop a
strength-of-materials-type solution for curved beams from the assumption that
plane sections remain plain during flexure. We can then compare this Winkler
solution* to the more accurate results from elasticity theory to see when added
refinement is necessary.

Consider a differential
slice from a curved beam and call the tangent to the neutral axis at any
cross-section *x*, the radial coordinate, *y*, positive inward as
shown in Figure 6.4. Assume that:

a. The
transverse loading as well as the cross-section is symmetric and the bending
moment is in the plane of symmetry (the *xy* plane).

b. The
radial stress is neglected.

c. If
there is shear and/or normal force on the cross-section they induce shear
and/or normal stresses as in the elementary straight beam formulas.

d. Transverse
sections remain plane (i.e., plane *câ€'c* rotates around the *z *axis
an angle* *Î'dÎ¨)

While these are the
same assumptions used for the elementary analysis of straight beams, for a
curved beam the longitudinal fibers are not all the same original base length.
Therefore, linear displacements produce nonlinear strains. For a layer of
longitudinal fibers *n*â€'*n*, a distance *y* from the neutral
axis at radius *r*, the rotation of the cross-section Î'dÎ¨ due to M gives:

which is hyperbolic,
not linear. For equilibrium, the neutral axis must be below the centroidal axis
(radius *R*) such that:

Thus for a curved beam,
the formulas for flexure with *N* and *V* by this Winkler analysis
are:

where *e*, the
distance from the centroidal axis to the neutral axis, is a property of the
cross-section such that (b) is satisfied. Formulas for various useful shapes
are given in many books. For the rectangular cross-section of unit thickness
necessary for comparison to an elasticity solution:

showing that only for
very deep beams is there an appreciable increase in the flexural stiffness.

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Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : Simplified Analysis of Curved Beams |

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