Consider the semicircular beam of rectangular cross-section bent by radial loads P applied at either end.

**Circular Beams
with End Loads **

Consider the semicircular beam of rectangular
cross-section bent by radial loads *P *applied at either end as shown in
Figure 6.7. The cross-section is rectangular* *of unit width and assume
plane stress. The bending moment at any radial cut is *PR *sinÎ¸* *and
from the Winkler solution we would expect that* ** *ÏƒÎ¸*f*(*r*)sinÎ¸.*
*Therefore from Section 6.4 we need a stress function of Type II with *m = *1.

For this exact solution
to apply at the ends of the beam, *P* must be distributed as dictated by
the solution for Ï„* _{r}*Î¸ at Î¸ = 0 and Ï€. If this is not the case, the solution
will, by St. Venantâ€™s principle, still apply at distances away from the end in
the order of (

Detailed
comparison to the simpler Winkler solution [Equation (6.19)] is left to the
chapter problems, but the agreement is again extremely close. The

The elastic rotation, Ï‰* _{z}*,
satisfying Laplaceâ€™s equation can be found directly from the Cauchy-Riemann
conditions. The first stress invariant is:

This is roughly the same relative difference as it
was for pure bending.

The elastic displacement field can be determined by
integrating the strains in the same way as was done for pure bending. Following
this procedure:

where the constants *H*, *K*, and *L*
are calculated from the geometric conditions associated with the supports. If
we assume the centerline is pinned at

corresponding to elementary analysis. Further
comparison of the deflections of thick rings is left to the solved problems,
but again the results from elas-ticity and Winkler analysis are in close
agreement.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

Civil : Principles of Solid Mechanics : Two Dimensional Solutions for Straight and Circular Beams : Circular Beams with End Loads |

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