Problem 10.26 by Castigliano II

Determine the bending moment and the force in the ring at point A.  Take into account the bending strain energy only.

Symmetry allows to consider a half ring with end conditions as shown. The lateral displacement and the rotation at point A are zero. Thus, by Castigliano's theorem

∂U*/∂N_A = 0

∂U*/∂M_A = 0

 By the method of sections, we have

For, 0 < q < p/2,    M₍₁₎ = M_A + R(1-cosq)N_A -PRsinq/2

For, p/2 < q < p,    M₍₂₎ = M_A + R(1-cosq)N_A-PR/2

Since only the bending strain energy is considered, we have for a thin ring,

U* = ∫M²Rdq/2EI

and

M₍₁₎² = M_A² + R²(1-cosq)²N_A² + 2RM_AN_A(1-cosq) -PRM_Asinq - PR²N_Asinq(1-cosq) + P²-term

M₍₂₎² = M_A² + R²(1-cosq)²N_A² + 2RM_AN_A(1-cosq) -PRM_A - PR²N_A(1-cosq) + P²-term

Using the following integrals

we obtain

U* is proportional to the above expression.  Setting to zero the partial derivatives with respect to N_A and M_A yields

The solution is