Problem 5.37 - By the differential equation method for determining the displacements.

Derive the stress formula

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To obtain N and M as functions of q, isolate the part of the ring above the horizontal diameter.  The shear forces at A and B are zero because line AB is an axis of symmetry of the ring.  The half ring is symmetric with respect to the vertical diameter.  Thus the  bending moments at A and B are equal, and the normal forces are equal to P/2, as shown.  From the second free-body diagram we obtain,

for 0 < q < p/2,

N = -(P/2)cosq

M = M_B - (P/2)R(1-cosq)

The formula to be established is the general stress formula (Text: Eq. 5.72, Notes: eq. 24, Sec.2.2) in which N and M are the expressions found above, with

M_B = M_A = 0.182PR

The problem is reduced to determining M_B or M_A and finding it as stated above.  M_B is a statical redundant, which is to be determined by the differential equation method for the displacements.  To do so, the half-ring on the left side of the vertical diameter is chosen for analysis.  By symmetry with respect to the vertical diameter, the rotations and horizontal displacements at C and D are zero.  Thus D is fixed, and the conditions at C are represented by a guided roller.  It is noted that the reaction N_o at C is equal to the shear force at A, and this was found above to be zero.  Instead of implementing this in the equations, we will find it as part of the solution.  The displacements of a half ring fixed at one end, and subjected to forces and a moment at the other end are determined (Notes, Sec. 4, Example 1) in the form of the flexibility relations

In the present problem,

V_o = P/2

v_o = 0

j_o = 0

The second and third flexibility equations yield, after dividing through, respectively, by R² and R,

- RP - pR(3/2 + Z)N_o  - p(1+Z)M_o = 0

- RP - pR(1 + Z)N_o  - p(1+Z)M_o = 0

The solution of these equation is

N_o = 0

M_o = - RP/p(1+Z)

From the figure above, the bending moment at A, M_A, is computed as

M_A = M_o + PR/2 = PR[1/2 - 1/p(1+Z)]

To find the desired result, M_A = 0.182PR, we need to neglect Z with respect to 1.  The result is then

M_A = PR[1/2 - 1/p] = 0.1817PR ≈ 0.182PR